-illion converter Translator

TREE(3) SSCG(3) Grand pentacthulhum · {100, 3, 2 [1 [1 ¬ 3] 2] 2} Pentacthulcross · {100, 100 [1 [1 ¬ 3] 1 \ 2] 2} Kingulus · {100, 100 [1 [1 ¬ 3] 1 [1 [1 ¬ 3] 2] 2] 2} Quadrunculus · {10, 100 [1 [1 ¬ 3] 1 [1 ¬ 3] 2] 2} Hexacthulhum· {100, 100 [1 [1 ¬ 3] 1 [1 ¬ 3] 2] 2} Godsgodgulus · {100, 99 [1 [2 ¬ 3] 2] 2} {100, 99 [1 [1, 2 ¬ 3] 2] 2} {100, 99 [1 [1 \ 2 ¬ 3] 2] 2} {100, 99 [1 [1 ¬ 4] 1, 1, 2] 2} Agoraphobia · {100, 100 [1 [1 ¬ 4] 1 \ 2] 2} {100, 100 [1 [1 ¬ 4] 1 \ 1 \ 2] 2} {100, 100 [1 [1 ¬ 5] 2] 2} Demagogue · {100, 99 [1 [1 ¬ 10] 2] 2} · E100{#, 10 (1) 2}100 (ExE, with BEAF structures), E100{#, 10 [2] 2}100 (ExE with Bird’s array notation structures) Ominongulus · {100, 100 [1 [1 ¬ 1, 2] 2] 2} · E100{#, 100 (1) 2}100 (ExE With BEAF structures), E100{#, 100 [2] 2}100 (ExE With Bird’s array notation structures) Pseudomonarchia daemonum · {10, 100 [1 [1 ¬ 44435623] 2] 2} · E100{#, # (1) 2}44,435,622 (ExE With BEAF structures), E100{#, # [2] 2}44,435,622 (ExE With Bird’s array notation structures) Extremebixul · 200![1(1)[2200, 200, 200, 200, 200]] · {200, 200 [1 [1 ¬ 1 ¬ 2] 2] 2} · {X, X (1) X^2} & 200 Kiloextremebixul · (200![1(1)[2200, 200, 200, 200, 200]])![1(1)[2200, 200, 200, 200, 200]] · {200, 200 [1 [1 ¬ 1 ¬ 1 [1 [1 ¬ 1 ¬ 2] 2] 2] 2] 2} Extremetrixul · 200![1(1)[2200, 200, 200, 200, 200, 200]] · {200, 200 [1 [1 ¬ 1 ¬ 1 ¬ 2] 2] 2} · {X, X (1) X^3} & 200 Extremequaxul · 200![1(1)[2200, 200, 200, 200, 200, 200, 200]] · {200, 200 [1 [1 ¬ 1 ¬ 1 ¬ 1 ¬ 2] 2] 2} · {X, X (1) X^4} & 200 … Big Boowa = \lbrace 3,3,3 / 2 \rbrace = f_{\psi_0(\Omega_{\omega}) + 2} (3) Big Hoss = \{100,100 \underbrace{////\ldots////}_{100} 2\} = \{100,100 (1)/ 2\} = f_{\psi_0(\Lambda)}(100) = f_{\psi_\Omega(\psi_I(0))}(100) Great Big Hoss = \{\text{big hoss},\text{big hoss} \underbrace{////\ldots////}_{\text{big hoss}} 2\} = f_{\psi_0(\Lambda)}^2(100) Bukuwaha = \{L,X^X\}_{100,100} BIGG = 200? = 200![[_{<1(200)2>⁅200⁆} 1]] = f_{\psi(\psi_{I_{\omega}}(0))}(200) Goshomity = \{100,100 \underbrace{\backslash\backslash\backslash\backslash\ldots\backslash\backslash\backslash\backslash}_{100} 2\} = \lbrace L2,100\rbrace_{100,100} Meameamealokopoowa = \{L100,10\}_{10,10} = f_{C(C_1(\Omega)99+9)}(10) Meameamealokopoowa-Oompa = \{\{L100,10\}_{10,10}\&L,10\}_{10,10} Tritar = Tar(3) Quadritar = Tar(4) Quintitar = Tar(5) ... Dekotar = Tar(10) Hektotar = Tar(100) Kilotar/Chilliotar = Tar(1 000) ... Unintar = Tar(Tar(10)) Bintar = Tar(Tar(Tar(10))) Trintar = Tar^4(10) ... Tarintar = Tar^{Tar}(Tar) Bashicu-matrix number with respect to Bashicu matrix system = (0,0,0,....)(1,1,1…)(2,2,2,....)....[n] Loader's Number = D^5(99) Yudkowsky’s number ６ (N primitive) Y sequence number = f^{2000}(1) ω-Y sequence number 1919th Busy Beaver = Σ(1919) Fish Number 1 Fish number 2 Fish Number 3 Fish Number 4 = F_4^{63}(3) 10^6th Xi function number = Ξ(1 000 000) \Sigma_\infty(10^9) Fish Number 5 Fish Number 6 Rayo's Number Fish number 7 = F_{7}^{63}(10^{100}) Fish number 8 Fish number 9 Fish number 10 Fish number 100 … Puny Foot · FOOT(1) Tiny Foot · FOOT(10) Small Foot · FOOT(10^10) Little Foot · FOOT^5(10^25) Foot · FOOT^10(10^50) BIG FOOT = \text{FOOT}^{10}(10^{100}) Bigger Foot · FOOT^10(10^1000000) Biggest Foot · FOOT^10(10^10^100) Large Foot · FOOT^10(10^10^1000000) Larger Foot · FOOT^10(10^^100) Largest Foot · FOOT^10(10^^^100) Huge Foot · FOOT^10(10~100) Huger Foot · FOOT^10(10~1000) Hugest Foot · FOOT^10(10~10000) Enormous Foot · FOOT^10(10~10~10) Ginormous Foot · FOOT^10(10{10}10) Little Bigeddon Sasquatch Double Sasquatch Triple Sasquatch … Large Garden Number = f^{10}(10 \uparrow^{10} 10) Hollom's number = \approx \mathrm{I}_{6.895 \times 10^{104}}(200) Oblivion = K(gongulus) Utter Oblivion = the largest finite number that can be uniquely defined using no more than an oblivion symbols Ultimate Oblivion Hyper Oblivion Sam's Number = \mathrm{Sam}_{\omega_{\varepsilon_{\omega_1^{\mathrm{CK}}\times f_{\psi(K)\times \mathrm{E}100\#100\#100}(\mathrm{Rayo}(10^{100}))}}^{\mathrm{CK}}}(Large Garden Number) Crouton Croutonillion Finity Infinity = ∞ or ∞ Aleph Null Beth Null ω ω+1 ω2 ω^2 ω^ω ε0 ε1 ζ0 η0 φ_4(0) φ_ω(0) Γ0 = Fefermann Schutte Ordinal β_0 = φ(1,1,0) δ_{0} = φ(1,2,0) λ_{0} = φ(1,3,0) α_{0} = φ(1,4,0) ψ(Ω^Ω^2) = φ(1,0,0,0) = Ackermann Ordinal ψ(ε_{Ω+1}) = Bachmann Howard Ordinal ψ(ε_{Ω_{ω}+1}}) = Takeuti Feferman Buchholz's Ordinal ψ(Ω_{ω+1}) = an ordinal greater than Takeuti Feferman Buchholz's Ordinal ψ(ψ_Ι(0)) = ψ(Ω_Ω_Ω_Ω_Ω_...) = ψ(Λ) = Extended Buchholz's Ordinal = Countable limit of the Εxtended Buchholz’s function \psi_{\Omega}(\varepsilon_{I+1}) with respect to the inaccessible ψ function \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) = Small Rathjen Ordinal \Psi^0_\Omega(\varepsilon_{K+1}) in Stegert’s psi Limit of Taranovsky’s function \omega_1^{\mathfrak{Ch}} = Omega one of chess \omega_1^{\mathfrak{Ch}_3} = Omega one of checkers ω₁ᶜᴷ = Church Kleene Ordinal ω_1 = Ω = ℵ_1 = First Uncountable Ordinal Omega One / Aleph One Ω_Ω_Ω_Ω_Ω_.... = ψ_Ι(0) = ℵ_ℵ_ℵ_... = ω_ω_ω_... = Λ = Omega Fixed Point / First Aleph Fixed Point Beth One Beth One · ℶ1 · Also Defined as the Cardinality of the Continuum Beth Two · ℶ2 Beth Omega · בω Gimel One · ℷ1 Gimel Two · ℷ2 Dalet One · ℸ1 He One · ה Vav One · Zayin One · Het One · Tet One · Yod One · Final Kaf One · Kaf One · Lamed One · Final Mem One · Mem One · Final Nun One · Nun One · Samekh One · Ayin One · Final Pe One · Pe One · Final Tsadi One · Tsadi One · Qof One · Resh One · Shin One · Tav One · Gamma Cardinal Theta Cardinal I = Inacessible Cardinal M = Mahlo Cardinal K = Weakly Compact Cardinal Ξ = Π₀²-indescribable cardinal Π = Indescribable Cardinal i = Rank-into Rank Cardinal 0=1 = Zero equal one cardinal \color{red}{\Omega} = ABSOLUTE INFINITY!!! The standard names follow this pattern (short scale): - 10^6: million - 10^9: billion - 10^12: trillion - 10^15: quadrillion - 10^33: decillion - 10^36: undecillion - 10^63: vigintillion - 10^66: unvigintillion - 10^303: centillion - 10^306: uncentillion - 10^309: duocentillion - 10^312: trescentillion Examples: - Input: "1e6" -> "1 million" - Input: "1.5e7" -> "15 million" - Input: "2e9" -> "2 billion" - Input: "3.45e12" -> "3.45 trillion" - Input: "2^20" -> "1.049 million" - Input: "10^304" -> "10 centillion" - Input: "1 023" -> "1 023 (using spaces)"). **Standard Abbreviations:** - 10^6: M (Million) - 10^9: B (Billion) - 10^12: T (Trillion) - 10^15: Qa or q (Quadrillion) - 10^18: Qi or Q (Quintillion) - 10^21: Sx or s (Sextillion) - 10^24: Sp or S (Septillion) - 10^27: Oc or O (Octillion) - 10^30: No or N (Nonillion) - 10^33: Dc or D (Decillion) - 10^36: UDc or UD (Undecillion) - 10^39: DDc or DD (Duodecillion) - 10^42: TDc or TD (Tredecillion) - 10^45: qDc or qD (Quattordecillion) - 10^48: QDc or QD (Quinquadecillion) … - 10^63: Vg or V (Vigintillion) - 10^66: UV or UVg (Unvigintillion) - 10^303: Ce or C (Centillion) - 10^603: Duc or Du (Ducentillion) - 10^903: Tc or Te (Trecentillion/Trucentillion) - 10^3 003: Mi (Millillion/Millinillion) **Beyond Millillion (Tier System based on N value where E = 3N + 3) Start of Tier 2:** **Millillion-based (N in thousands):** - N=1000 (millillion/millinillion): Mi - N=1001 (milli-unillion): Mi-U - N=1002 (milli-duillion): Mi-D - N=1010 (milli-decillion): Mi-De - N=1100 (milli-centillion): Mi-Ce - N=2000 (dumillillion): DMi - N=3000 (trimillillion): TMi - N=10 000 (decimillillion): DeMi - N=20 000 (vigintimillillion): VgMi - N=100 000 (centimillillion): CeMi **Myrillion-based (alternative naming):** - N=10 000 (myrillion): My - N=11 000 (myrimillillion): My-Mi - N=20 000 (dumyrillion): DMy - N=100 000 (decimyrillion): DeMy **Higher Tier Prefixes (based on N):** - N=10^6 (micrillion): Mc - N=10^6+1 (micro-unillion): Mc-U - N=10^6+1000 (micro-millillion): Mc-Mi - N=2*10^6 (dumicrillion): DMc - N=10^9 (nanillion): Na - N=2*10^9 (dunanillion): DNa - N=10^12 (picillion): Pc - N=10^15 (femtillion): Fem - N=10^18 (attillion): At - N=10^21 (zeptillion): Zep - N=10^24 (yoctillion): Yo - N=10^27 (rontillion): Rt - N=10^30 (quectillion): Qu - N=10^33 (mecillion): Me(c) - N=10^36 (duecillion): Du(c) - N=10^39 (trecillion): Tre(c) - N=10^42 (tetrecillion): Te(c) - N=10^45 (pentecillion): Pe(c) - N=10^48 (hexecillion): He(c) - N=10^51 (heptecillion): Hp(c) - N=10^54 (octecillion): Ot(c) - N=10^57 (ennecillion): En(c) - N=10^60 (icosillion): Is(o) - N=10^90 (triacontillion): TrC - N=10^120 (tetracontillion): TeC - N=10^150 (pentacontillion): PeC - N=10^180 (hexacontillion): HeC - N=10^210 (heptacontillion): HpC - N=10^240 (octacontillion): OtC - N=10^270 (enneacontillion): EnC - N=10^300 (hectillion): Hec - N=10^600 (dohectillion): DHc - N=10^900 (triahectillion): TrH etc. The tier-2 limit and tier-3 starts at: - N=10^3000 (killillion): Kl. Use AAS (Aarex’s Abbreviation System) for Abbreviations instead: “Like Standard, but the parts are changed. Also in standard, there is a space after the number then you have the abbreviation for it (ex: 10 Qa, 123.45 Sp). However in AAS, the space is after the number (ex: 10 Qa, 123.45 Sp). Unit parts: K, M, B, T, Qa, Qi, Sx, Sp, Oc, and No. Units place: U, D, T, Qt, Qn, Sx, Sp, O, N Tens place: D/De, V/Vg, Tg, Qd, Qq, Sc, St, Oe, Ng Hundreds place: C/Ce, Dc, Tc, Qe, Qu, Se, Si, Oe, Ne Special roots: Mi, Mc, Na, Pc, Fem, At, Zep, Yo, etc. D and De are toggled using the notation options. How to express, calculate equate, etc. it. (e.g. \begin{aligned}2\uparrow \uparrow \uparrow 4&=H_{5}(2,4)&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow 2))&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow 2))&=2\uparrow \uparrow (2\uparrow \uparrow 4)&=\underbrace {2\uparrow (2\uparrow (2\uparrow \cdots ))} ;=;\underbrace {;2^{2^{\cdots ^{2}}}} &;;;;;2\uparrow \uparrow 4{\text{ copies of }}2;;;;;{\text{65 536 2s}}\end{aligned}) \begin{eqnarray*} f_0(n) &=& n + 1 \\ f_1(n) &=& f_0^n(n) = ( \cdots ((n + 1) + 1) + \cdots + 1) = n + n = 2n \\ f_2(n) &=& f_1^n(n) = 2(2(\ldots 2(2n))) = 2^n n > 2^n \\ f_3(n) &>& 2\uparrow(2\uparrow(\ldots 2\uparrow(2\uparrow n))) > 2\uparrow\uparrow n\\ f_4(n) &>& 2\uparrow\uparrow(2\uparrow\uparrow(\ldots 2\uparrow\uparrow n)) > 2\uparrow\uparrow\uparrow n \\ f_m(n) &>& 2\uparrow^{m-1} n \\ f_\omega(n) &=& f_n(n) > 2\uparrow^{n-1} n > \text{Ack}(n) \\ f_{\omega+1}(n) &>& \lbrace n,n,1,2 \rbrace \\ f_{\omega+2}(n) &>& \lbrace n,n,2,2 \rbrace \\ f_{\omega+m}(n) &>& \lbrace n,n,m,2 \rbrace \\ f_{\omega\times 2}(n) &>& \lbrace n,n,n,2 \rbrace \\ f_{\omega\times 3}(n) &>& \lbrace n,n,n,3 \rbrace \\ f_{\omega\times m}(n) &>& \lbrace n,n,n,m \rbrace \\ f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace \\ f_{\omega^m}(n) &>& \lbrace n,m+2 [2] 2 \rbrace \\ f_{\omega^{\omega}}(n) &>& \lbrace n,n+2 [2] 2 \rbrace > \lbrace n,n [2] 2 \rbrace \\ f_{\omega^{\omega}+1}(n) &>& \lbrace n,n,2 [2] 2 \rbrace \\ f_{\omega^{\omega}+2}(n) &>& \lbrace n,n,3 [2] 2 \rbrace \\ f_{\omega^{\omega}+m}(n) &>& \lbrace n,n,m+1 [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega}(n) &>& \lbrace n,n,n+1 [2] 2 \rbrace > \lbrace n,n,n [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega+1}(n) &>& \lbrace n,n,1,2 [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega2}(n) &>& \lbrace n,n,n,2 [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega^2}(n) &>& \lbrace n,n,n,n [2] 2 \rbrace \\ f_{{\omega^{\omega}}2}(n) &>& \lbrace n,n [2] 3 \rbrace \\ f_{{\omega^{\omega}}3}(n) &>& \lbrace n,n [2] 4 \rbrace \\ f_{{\omega^{\omega}}m}(n) &>& \lbrace n,n [2] m+1 \rbrace \\ f_{\omega^{\omega+1}}(n) &>& \lbrace n,n [2] n+1 \rbrace > \lbrace n,n [2] n \rbrace \\ f_{\omega^{\omega+2}}(n) &>& \lbrace n,n [2] n,n \rbrace \\ f_{\omega^{\omega+3}}(n) &>& \lbrace n,n,n [2] n,n,n \rbrace \\ f_{\omega^{\omega+m}}(n) &>& \lbrace n,m [2] 1 [2] 2 \rbrace \\ f_{\omega^{\omega2}}(n) &>& \lbrace n,n [2] 1 [2] 2 \rbrace > \lbrace n,2 [3] 2 \rbrace \\ f_{\omega^{\omega3}}(n) &>& \lbrace n,n [2] 1 [2] 1 [2] 2 \rbrace > \lbrace n,3 [3] 2 \rbrace \\ f_{\omega^{\omega m}}(n) &>& \lbrace n,m [3] 2 \rbrace \\ f_{\omega^{\omega^2}}(n) &>& \lbrace n,n [3] 2 \rbrace \\ f_{\omega^{\omega^3}}(n) &>& \lbrace n,n [4] 2 \rbrace \\ f_{\omega^{\omega^m}}(n) &>& \lbrace n,n [m+1] 2 \rbrace \\ f_{\omega^{\omega^\omega}}(n) &>& \lbrace n,n [n+1] 2 \rbrace > \lbrace n,n [1,2] 2 \rbrace \\ f_{^4{\omega}}(n) &>& \lbrace n,n [1 [2] 2] 2 \rbrace \\ f_{^5{\omega}}(n) &>& \lbrace n,n [1 [1,2] 2] 2 \rbrace \\ f_{^6{\omega}}(n) &>& \lbrace n,n [1 [1 [2] 2] 2] 2 \rbrace \\ f_{\varepsilon_0}(n) &>& \lbrace n,n [1 / 2] 2 \rbrace \\ f_{\varepsilon_02}(n) &>& \lbrace n,n [1 / 2] 3 \rbrace \\ f_{\varepsilon_03}(n) &>& \lbrace n,n [1 / 2] 4 \rbrace \\ f_{\varepsilon_0m}(n) &>& \lbrace n,n [1 / 2] m+1 \rbrace \\ f_{\varepsilon_0\omega}(n) &>& \lbrace n,n [1 / 2] n+1 \rbrace \\ f_{\varepsilon_0{\omega^{\omega}}}(n) &>& \lbrace n,n [1 / 2] 1 [2] 2 \rbrace \\ f_{\varepsilon_0{\omega^{\omega^{\omega}}}}(n) &>& \lbrace n,n [1 / 2] 1 [1,2] 2 \rbrace \\ f_{\varepsilon_0{\omega^{\omega^{\omega^{\omega}}}}}(n) &>& \lbrace n,n [1 / 2] 1 [1 [2] 2] 2 \rbrace \\ f_{\varepsilon_0^2}(n) &>& \lbrace n,n [1 / 2] 1 [1 / 2] 2 \rbrace \\ f_{\varepsilon_0^3}(n) &>& \lbrace n,n [1 / 2] 1 [1 / 2] 1 [1 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\omega}}(n) &>& \lbrace n,n [2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\omega^{\omega}}}(n) &>& \lbrace n,n [1,2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\omega^{\omega^{\omega}}}}(n) &>& \lbrace n,n [1 [2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\varepsilon_0}}(n) &>& \lbrace n,n [1 [1 / 2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(n) &>& \lbrace n,n [1 [1 / 2] 1 [1 / 2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(n) &>& \lbrace n,n [1 [1 [1 / 2] 2 / 2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_1}(n) &>& \lbrace n,n [1 / 3] 2 \rbrace \\ f_{\varepsilon_2}(n) &>& \lbrace n,n [1 / 4] 2 \rbrace \\ f_{\varepsilon_{\omega}}(n) &>& \lbrace n,n [1 / 1,2] 2 \rbrace \\ f_{\varepsilon_{\omega^2}}(n) &>& \lbrace n,n [1 / 1,1,2] 2 \rbrace \\ f_{\varepsilon_{\omega^{\omega}}}(n) &>& \lbrace n,n [1 / 1 [2] 2] 2 \rbrace \\ f_{\varepsilon_{\omega^{\omega^{\omega}}}}(n) &>& \lbrace n,n [1 / 1 [1,2] 2] 2 \rbrace \\ f_{\varepsilon_{\varepsilon_0}}(n) &>& \lbrace n,n [1 / 1 [1 / 2] 2] 2 \rbrace \\ f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}(n) &>& \lbrace n,n [1 / 1 [1 / 1 [1 / 2] 2] 2] 2 \rbrace \\ f_{\zeta_0}(n) &>& \lbrace n,n [1 / 1 / 2] 2 \rbrace \\ f_{\zeta_0^{\zeta_0}}(n) &>& \lbrace n,n [1 [1 / 1 / 2] 2 / 1 / 2] 2 \rbrace \\ f_{\varepsilon_{\zeta_0+1}}(n) &>& \lbrace n,n [1 / 2 / 2] 2 \rbrace \\ f_{\varepsilon_{\zeta_0+2}}(n) &>& \lbrace n,n [1 / 3 / 2] 2 \rbrace \\ f_{\varepsilon_{\varepsilon_{\zeta_0+1}}}(n) &>& \lbrace n,n [1 / 1 [1 / 2 / 2] 2 / 2]2 \rbrace \\ f_{\zeta_1}(n) &>& \lbrace n,n [1 / 1 / 3] 2 \rbrace \\ f_{\zeta_2}(n) &>& \lbrace n,n [1 / 1 / 4] 2 \rbrace \\ f_{\zeta_{\zeta_0}}(n) &>& \lbrace n,n [1 / 1 / 1 [1 / 1 / 2] 2] 2 \rbrace \\ f_{\eta_0}(n) &>& \lbrace n,n [1 / 1 / 1 / 2] 2 \rbrace \\ f_{\varphi(4,0)}(n) &>& \lbrace n,n [1 / 1 / 1 / 1 / 2] 2 \rbrace \\ f_{\varphi(\omega,0)}(n) &>& \lbrace n,n [1 [2 \sim 2] 2] 2 \rbrace \\ f_{\varphi(\varphi(\omega,0),0)}(n) &>& \lbrace n,n [1 [1 [1 [2 \sim 2] 2] 2 \sim 2] 2] 2 \rbrace \\ f_{\Gamma_0}(n) &>& \lbrace n,n [1 [1 / 2 \sim 2] 2] 2 \rbrace \\ f_{\varphi(1,0,0,0)}(n) &>& \lbrace n,n [1 [1 / 3 \sim 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega^{\Omega^{\omega}})}(n) &>& \lbrace n,n [1 [1 / 1, 2 \sim 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega^{\Omega^{\Omega}})}(n) &>& \lbrace n,n [1 [1 / 1 / 2 \sim 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega^{\Omega^{\Omega^{\Omega}}})}(n) &>& \lbrace n,n [1 [1 [1 / 2 \sim 2] 2 \sim 2] 2] 2 \rbrace \\ f_{\psi(\psi_1(1))}(n) &>& \lbrace n,n [1 [1 \sim 3] 2] 2 \rbrace \\ f_{\psi(\psi_1(2))}(n) &>& \lbrace n,n [1 [1 \sim 1 \sim 2] 2] 2 \rbrace \\ f_{\psi(\psi_1(\omega))}(n) &>& \lbrace n,n [1 [1 [2 /_3 2] 2] 2] 2 \rbrace \\ f_{\psi(\psi_1(\Omega))}(n) &>& \lbrace n,n [1 [1 [1 / 2 /_3 2] 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_2)}(n) &>& \lbrace n,n [1 [1 [1 \sim 2 /_3 2] 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_3)}(n) &>& \lbrace n,n [1 [1 [1 [1 /_3 2 /_4 2] 2] 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\omega})}(n) &>& \lbrace n,n [1 [2 /_{1,2} 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\varepsilon_0})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 / 2] 2} 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\Gamma_0})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 / 2 \sim 2] 2} 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\psi_0(\Omega_2)})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 [1 [1 \sim2 /_3 2] 2] 2] 2} 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\psi_0(\Omega_3)})}(n) &>& \lbrace n,n [1 [2/_{1 [1 [1 [1 [1 /_3 2/_4 2] 2] 2] 2] 2} 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\psi_0(\Omega_{\omega})})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 [2 /_{1,2} 2] 2] 2} 2] 2] 2 \rbrace \\ f_{\psi_0(\Omega_{\psi_0(\Omega_{\psi_0(\Omega_{\omega})})})}(n) &>& \lbrace n,n [1 [2 /_{1 [2 /_{1 [1 [2 /_{1,2} 2] 2] 2} 2] 2} 2] 2] 2 \rbrace \end{eqnarray*} S = {0,1,ω,Ω} C(0) = {Finite AME 0,1,ω,Ω) ψ(x) = smallest ordinal not a member of C(x) ω+1, ωx2, ω^ω, .....(infinite),..... = ε_0 = ψ(0) ψ(1) = ε_1 ψ(ω) = ε_ω ψ(ε_ε_ε_...) = ψ(ψ(ψ(...(...)))))....))) = ψ(ζ_0) = ψ(Ω) = ζ_0 ψ(Ω+1) = ε_{ζ_{0}+1}} ψ(Ω+2) = ε_{ζ_{0}+2}} ψ(Ω+a) = ε_{ζ_{0}+a}} ψ(Ω2) = ζ_1 ψ(Ω2) = ζ_2 ψ(Ωω) = ζ_ω ψ(Ω^2)= η_0 ψ(Ω^3)= φ_{4}(0) = φ(4,0) ψ(Ω^ω)= φ_{ω}(0) = φ(ω,0) ψ(Ω^Ω) = Γ_0 = φ(1,0,0) ψ(Ω^{Ω}a) = Γ_{a-1} ψ(Ω^{Ω+1}) = φ(1,1,0) ψ(Ω^{Ω+a}) = φ(1,a,0) ψ(Ω^{Ωa}) = φ(a,0,0) ψ(Ω^Ω^2) = φ(1,0,0,0) ψ(Ω^Ω^3) = φ(1,0,0,0,0) ψ(Ω^Ω^ω) = φ(1,0,0,0,0,0,0,......) (ω times)= SVO = φ(1 @^ω) ψ(Ω^Ω^Ω) = φ(1,0,0,0,0,0,0,......) (Ω times) = LVO = φ(1 @^{1,0}) We can continue exponentiating Ω's and more (e.g. φ(1 @^{1,0}, φ(1 @^{1 @^{1,0}}, etc.) like this, but only a finite number of times. If we do it an infinite number of times, then we'll reach our next ordinal which is.... ψ(Ω^Ω^Ω^...) = ψ(Ω↑↑ω) = ψ(ε_{Ω+1}) = ψ(ψ_1(0)) = ψ_1(0) = ψ(Ω_2) (in Buchholz's function, extended from Googology Wiki) = BHO \underbrace{C(C(\cdots(C(C(\Omega_{n}2,0),0),\cdots ),0)}_{n\text{ C's}} = Taranovsky's function ​​Use this format: "1000" (no spaces) (as 10^3), "10 000" (as 10^4), etc., “0.001” (as 10^-3), “0.000 1” (as 10^-4), “0.000 01” (as 10^-5), “0.000 001” (as 10^-6), etc. Example of a respect to an unspecified (and perhaps ill-defined) OCF \psi Ordinal List: 1, 2, 3, 4, 5, "$\\omega$", "$\\omega+1$", "$\\omega+2$", "$\\omega+3$", "$\\omega+4$", "$\\omega+5$", "$\\omega 2$", "$\\omega 3$", "$\\omega 4$", "$\\omega 5$", "$\\omega^2$", "$\\omega^3$", "$\\omega^4$", "$\\omega^5$", "$\\omega^\\omega$", "$\\omega^{\\omega^{\\omega}}$", "$\\omega^{\\omega^{\\omega^{\\omega}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\omega}}}}}$", "$\\varepsilon_0$", "$\\varepsilon_0+1$", "$\\varepsilon_0+2$", "$\\varepsilon_0+3$", "$\\varepsilon_0+4$", "$\\varepsilon_0+5$", "$\\varepsilon_0 2$", "$\\varepsilon_0 3$", "$\\varepsilon_0 4$", "$\\varepsilon_0 5$", "$\\omega^{\\varepsilon_0+1}$", "$\\omega^{\\varepsilon_0+1}+1$", "$\\omega^{\\varepsilon_0+1}2$", "$\\omega^{\\varepsilon_0+1}3$", "$\\omega^{\\varepsilon_0+2}$", "$\\omega^{\\varepsilon_0+3}$", "$\\omega^{\\varepsilon_0+4}$", "$\\omega^{\\varepsilon_0+5}$", "$\\omega^{\\varepsilon_0+5}$", "$\\omega^{\\varepsilon_0 2}$", "$\\omega^{\\varepsilon_0 3}$", "$\\omega^{\\varepsilon_0 4}$", "$\\omega^{\\varepsilon_0 5}$", "$\\omega^{\\omega^{\\varepsilon_0+1}}$", "$\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}}}}$", "$\\varepsilon_1$", "$\\varepsilon_1+1$", "$\\varepsilon_1+2$", "$\\varepsilon_1+3$", "$\\varepsilon_1+4$", "$\\varepsilon_1+5$", "$\\varepsilon_1+\\omega$", "$\\varepsilon_1+\\omega 2$", "$\\varepsilon_1+\\omega 3$", "$\\varepsilon_1+\\omega 4$", "$\\varepsilon_1+\\omega^{\\omega}$", "$\\varepsilon_1+\\omega^{\\omega^{\\omega}}$", "$\\varepsilon_1+\\varepsilon_0$", "$\\varepsilon_1+\\omega^{\\varepsilon_0}$", "$\\varepsilon_1+\\omega^{\\omega^{\\varepsilon_0}}$", "$\\varepsilon_1+\\omega^{\\omega^{\\omega^{\\varepsilon_0}}}$", "$\\varepsilon_1 2$", "$\\varepsilon_1 3$", "$\\varepsilon_1 4$", "$\\varepsilon_1 5$", "$\\omega^{\\varepsilon_1+1}$", "$\\omega^{\\varepsilon_1+2}$", "$\\omega^{\\varepsilon_1+3}$", "$\\omega^{\\varepsilon_1+4}$", "$\\omega^{\\varepsilon_1+5}$", "$\\omega^{\\varepsilon_1 2}$", "$\\omega^{\\varepsilon_1 3}$", "$\\omega^{\\varepsilon_1 4}$", "$\\omega^{\\varepsilon_1 5}$", "$\\omega^{\\omega^{\\varepsilon_1+1}}$", "$\\omega^{\\omega^{\\omega^{\\varepsilon_1+1}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_1+1}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_1+1}}}}}$", "$\\varepsilon_2$", "$\\varepsilon_3$", "$\\varepsilon_4$", "$\\varepsilon_5$", "$\\varepsilon_6$", "$\\varepsilon_\\omega$", "$\\varepsilon_{\\omega+1}$", "$\\varepsilon_{\\omega+2}$", "$\\varepsilon_{\\omega+3}$", "$\\varepsilon_{\\omega+4}$", "$\\varepsilon_{\\omega 2}$", "$\\varepsilon_{\\omega 3}$", "$\\varepsilon_{\\omega 4}$", "$\\varepsilon_{\\omega^{\\omega^\\omega}}$", "$\\varepsilon_{\\varepsilon_0}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}}}$", "$\\zeta_0$", "$\\varepsilon_{\\zeta_0}$", "$\\varepsilon_{\\varepsilon_{\\zeta_0}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\zeta_0}}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{\\zeta_0} }}}$", "$\\zeta_1$", "$\\zeta_2$", "$\\zeta_3$", "$\\zeta_4$", "$\\zeta_5$", "$\\zeta_\\omega$", "$\\zeta_{\\omega^2}$", "$\\zeta_{\\omega^3}$", "$\\zeta_{\\omega^4}$", "$\\zeta_{\\omega^\\omega}$", "$\\zeta_{\\omega^{\\omega^\\omega}}$", "$\\zeta_{\\varepsilon_{0}}$", "$\\zeta_{\\varepsilon_{1}}$", "$\\zeta_{\\varepsilon_{2}}$", "$\\zeta_{\\varepsilon_{3}}$", "$\\zeta_{\\varepsilon_{4}}$", "$\\zeta_{\\varepsilon_{\\varepsilon_{0}}}$", "$\\zeta_{\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}}$", "$\\zeta_{\\zeta_{0}}$", "$\\zeta_{\\zeta_{\\zeta_{0}}}$", "$\\zeta_{\\zeta_{\\zeta_{\\zeta_{0}}}}$", "$\\varphi_{3}(0)$", "$\\varphi_{3}(1)$", "$\\varphi_{3}(2)$", "$\\varphi_{3}(3)$", "$\\varphi_{3}(4)$", "$\\varphi_{3}(5)$", "$\\varphi_{3}(\\omega)$", "$\\varphi_{3}(\\varepsilon_0)$", "$\\varphi_{3}(\\zeta_0)$", "$\\varphi_{3}(\\varphi_{3}(0))$", "$\\varphi_{3}(\\varphi_{3}(\\varphi_{3}(0)))$", "$\\varphi_{3}(\\varphi_{3}(\\varphi_{3}(\\varphi_{3}(0))))$", "$\\varphi_{4}(0)$", "$\\varphi_{4}(1)$", "$\\varphi_{4}(2)$", "$\\varphi_{4}(3)$", "$\\varphi_{4}(\\omega)$", "$\\varphi_{4}(\\varepsilon_{0})$", "$\\varphi_{4}(\\zeta_{0})$", "$\\varphi_{4}(\\varphi_{3}(0))$", "$\\varphi_{4}(\\varphi_{4}(0))$", "$\\varphi_{5}(0)$", "$\\varphi_{6}(0)$", "$\\varphi_{7}(0)$", "$\\varphi_{8}(0)$", "$\\varphi_{\\omega}(0)$", "$\\varphi_{\\varepsilon_{0}}(0)$", "$\\varphi_{\\zeta_{0}}(0)$", "$\\varphi_{\\varphi_{3}(0)}(0)$", "$\\varphi_{\\varphi_{4}(0)}(0)$", "$\\varphi_{\\varphi_{5}(0)}(0)$", "$\\varphi_{\\varphi_{6}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)}(0)}(0)}(0)$", "$\\varphi(1,0,0)$", "$\\varphi(1,0,1)$", "$\\varphi(1,0,2)$", "$\\varphi(1,0,3)$", "$\\varphi(1,0,4)$", "$\\varphi(1,0,5)$", "$\\varphi(1,0,\\omega)$", "$\\varphi(1,0,\\varepsilon_{0})$", "$\\varphi(1,0,\\varphi_{2}(0))$", "$\\varphi(1,0,\\varphi_{3}(0))$", "$\\varphi(1,0,\\varphi_{4}(0))$", "$\\varphi(1,0,\\varphi_{5}(0))$", "$\\varphi(1,0,\\varphi_{\\omega}(0))$", "$\\varphi(1,0,\\varphi_{\\varepsilon_{0}}(0))$", "$\\varphi(1,0,\\varphi_{\\zeta_{0}}(0))$", "$\\varphi(1,0,\\varphi(1,0,0))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0)))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0)))))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))))))$", "$\\varphi(1,1,0)$", "$\\varphi(1,1,1)$", "$\\varphi(1,1,2)$", "$\\varphi(1,1,3)$", "$\\varphi(1,1,4)$", "$\\varphi(1,1,5)$", "$\\varphi(1,1,\\omega)$", "$\\varphi(1,1,\\varepsilon_{0})$", "$\\varphi(1,1,\\zeta_{0})$", "$\\varphi(1,2,0)$", "$\\varphi(1,2,1)$", "$\\varphi(1,2,2)$", "$\\varphi(1,2,3)$", "$\\varphi(1,2,4)$", "$\\varphi(1,2,5)$", "$\\varphi(1,3,0)$", "$\\varphi(1,4,0)$", "$\\varphi(1,5,0)$", "$\\varphi(1,6,0)$", "$\\varphi(1,7,0)$", "$\\varphi(1,8,0)$", "$\\varphi(1,\\omega,0)$", "$\\varphi(1,\\varepsilon_{0},0)$", "$\\varphi(1,\\zeta_{0},0)$", "$\\varphi(1,\\varphi_{3}(0),0)$", "$\\varphi(1,\\varphi_{\\omega}(0),0)$", "$\\varphi(1,\\varphi_{\\varepsilon_{0}}(0),0)$", "$\\varphi(1,\\varphi_{\\zeta_{0}}(0),0)$", "$\\varphi(1,\\varphi(1,0,0),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,0)),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0)))),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))))),0)$", "$\\varphi(2,0,0)$", "$\\varphi(2,0,1)$", "$\\varphi(2,0,2)$", "$\\varphi(2,0,3)$", "$\\varphi(2,0,4)$", "$\\varphi(2,0,5)$", "$\\varphi(2,1,0)$", "$\\varphi(2,2,0)$", "$\\varphi(2,3,0)$", "$\\varphi(2,4,0)$", "$\\varphi(2,5,0)$", "$\\varphi(2,6,0)$", "$\\varphi(3,0,0)$", "$\\varphi(4,0,0)$", "$\\varphi(5,0,0)$", "$\\varphi(6,0,0)$", "$\\varphi(7,0,0)$", "$\\varphi(8,0,0)$", "$\\varphi(1,0,0,0)$", "$\\varphi(1,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0,0,0,0)$", "$\\psi(\\Omega^{\\Omega^\\omega})$", "$\\psi(\\Omega^{\\Omega^\\omega+1})$", "$\\psi(\\Omega^{\\Omega^\\omega+2})$", "$\\psi(\\Omega^{\\Omega^\\omega+3})$", "$\\psi(\\Omega^{\\Omega^\\omega 2})$", "$\\psi(\\Omega^{\\Omega^\\omega 3})$", "$\\psi(\\Omega^{\\Omega^\\omega 4})$", "$\\psi(\\Omega^{\\Omega^\\omega 5})$", "$\\psi(\\Omega^{\\Omega^\\Omega})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+1}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+2}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+3}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+4}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+5}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 2}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 3}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 4}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 5}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{2}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{3}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{4}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{5}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\omega}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}}}}})$", "$\\psi(\\Omega_2)$", "$\\psi(\\Omega_2+1)$", "$\\psi(\\Omega_2+2)$", "$\\psi(\\Omega_2+3)$", "$\\psi(\\Omega_2+4)$", "$\\psi(\\Omega_2+5)$", "$\\psi(\\Omega_2+\\omega)$", "$\\psi(\\Omega_2+\\Omega)$", "$\\psi(\\Omega_2 2)$", "$\\psi(\\Omega_2 3)$", "$\\psi(\\Omega_2 4)$", "$\\psi(\\Omega_2 5)$", "$\\psi(\\Omega_2 \\omega)$", "$\\psi(\\Omega_2 \\Omega)$", "$\\psi(\\Omega_2^2)$", "$\\psi(\\Omega_2^3)$", "$\\psi(\\Omega_2^4)$", "$\\psi(\\Omega_2^5)$", "$\\psi(\\Omega_2^{\\Omega_2})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}}}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}}}}})$", "$\\psi(\\Omega_3)$", "$\\psi(\\Omega_3^{\\Omega_3})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3}})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3}}})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3}}}})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3}}}}})$", "$\\psi(\\Omega_4)$", "$\\psi(\\Omega_5)$", "$\\psi(\\Omega_6)$", "$\\psi(\\Omega_7)$", "$\\psi(\\Omega_8)$", "$\\psi(\\Omega_9)$", "$\\psi(\\Omega_\\omega)$", "$\\psi(\\Omega_\\Omega)$", "$\\psi(\\Omega_{\\Omega_{\\omega}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega}}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega}}}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega}}}}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega}}}}}})$", "$\\psi(\\psi_I(0))$", "$\\psi(\\psi_I(0)+1)$", "$\\psi(\\psi_I(0)+2)$", "$\\psi(\\psi_I(0)+3)$", "$\\psi(\\psi_I(0)+4)$", "$\\psi(\\psi_I(0)+5)$", "$\\psi(\\psi_I(0)2)$", "$\\psi(\\psi_I(0)3)$", "$\\psi(\\psi_I(0)4)$", "$\\psi(\\psi_I(0)5)$", "$\\psi(\\psi_I(0)^{\\psi_I(0)})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}}})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}}}})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}}}}})$", "$\\psi(\\psi_I(1))$", "$\\psi(\\psi_I(1)^{\\psi_I(1)})$", "$\\psi(\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)}})$", "$\\psi(\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)}}})$", "$\\psi(\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)}}}})$", "$\\psi(\\psi_I(3))$", "$\\psi(\\psi_I(4))$", "$\\psi(\\psi_I(5))$", "$\\psi(\\psi_I(6))$", "$\\psi(\\psi_I(\\omega))$", "$\\psi(\\psi_I(\\Omega))$", "$\\psi(\\psi_I(\\psi_I(0)))$", "$\\psi(\\psi_I(\\psi_I(\\psi_I(0))))$", "$\\psi(\\psi_I(\\psi_I(\\psi_I(\\psi_I(0)))))$", "$\\psi(\\psi_I(\\psi_I(\\psi_I(\\psi_I(\\psi_I(0))))))$", "$\\psi(I)$", "$\\psi(I+1)$", "$\\psi(I+2)$", "$\\psi(I+3)$", "$\\psi(I+4)$", "$\\psi(I+5)$", "$\\psi(I+\\omega)$", "$\\psi(I+\\Omega)$", "$\\psi(I+\\Omega_2)$", "$\\psi(I+\\Omega_3)$", "$\\psi(I+\\Omega_4)$", "$\\psi(I+\\Omega_5)$", "$\\psi(I+\\Omega_\\omega)$", "$\\psi(I+\\Omega_\\Omega)$", "$\\psi(I+\\Omega_{\\Omega_\\Omega})$", "$\\psi(I+\\psi_I(0))$", "$\\psi(I+\\psi_I(1))$", "$\\psi(I+\\psi_I(2))$", "$\\psi(I+\\psi_I(3))$", "$\\psi(I+\\psi_I(4))$", "$\\psi(I+\\psi_I(5))$", "$\\psi(I+\\psi_I(\\omega))$", "$\\psi(I+\\psi_I(\\Omega))$", "$\\psi(I+\\psi_I(\\psi_I(0)))$", "$\\psi(I+\\psi_I(\\psi_I(\\psi_I(0))))$", "$\\psi(I+\\psi_I(\\psi_I(\\psi_I(\\psi_I(0)))))$", "$\\psi(I2)$", "$\\psi(I3)$", "$\\psi(I4)$", "$\\psi(I5)$", "$\\psi(I6)$", "$\\psi(I\\omega)$", "$\\psi(I\\Omega)$", "$\\psi(I\\psi_I(0))$", "$\\psi(I\\psi_I(\\psi_I(0)))$", "$\\psi(I\\psi_I(\\psi_I(\\psi_I(0))))$", "$\\psi(I^2)$", "$\\psi(I^3)$", "$\\psi(I^4)$", "$\\psi(I^5)$", "$\\psi(I^6)$", "$\\psi(I^\\omega)$", "$\\psi(I^\\Omega)$", "$\\psi(I^{\\psi_I(0)})$", "$\\psi(I^{\\psi_I(1)})$", "$\\psi(I^{\\psi_I(2)})$", "$\\psi(I^{\\psi_I(3)})$", "$\\psi(I^{\\psi_I(4)})$", "$\\psi(I^{\\psi_I(\\omega)})$", "$\\psi(I^{\\psi_I(\\Omega)})$", "$\\psi(I^{\\psi(I)})$", "$\\psi(I^I)$", "$\\psi(I^{I^2})$", "$\\psi(I^{I^3})$", "$\\psi(I^{I^4})$", "$\\psi(I^{I^5})$", "$\\psi(I^{I^I})$", "$\\psi(I^{I^{I^I}})$", "$\\psi(I^{I^{I^{I^{I}}}})$", "$\\psi(I^{I^{I^{I^{I^{I}}}}})$", "$\\psi(I^{I^{I^{I^{I^{I^{I}}}}}})$", "$\\psi(\\psi_{I_2}(0))$", "$\\psi(\\psi_{I_2}(1))$", "$\\psi(\\psi_{I_2}(2))$", "$\\psi(\\psi_{I_2}(3))$", "$\\psi(\\psi_{I_2}(4))$", "$\\psi(\\psi_{I_2}(5))$", "$\\psi(I_2)$", "$\\psi(I_2+1)$", "$\\psi(I_2+2)$", "$\\psi(I_2+3)$", "$\\psi(I_2+4)$", "$\\psi(I_2+5)$", "$\\psi(I_2 2)$", "$\\psi(I_2 3)$", "$\\psi(I_2 4)$", "$\\psi(I_2 5)$", "$\\psi(I_2^{I_2})$", "$\\psi(I_2^{I_2^{I_2}})$", "$\\psi(I_2^{I_2^{I_2^{I_2}}})$", "$\\psi(I_2^{I_2^{I_2^{I_2^{I_2}}}})$", "$\\psi(I_3)$", "$\\psi(I_4)$", "$\\psi(I_5)$", "$\\psi(I_6)$", "$\\psi(I_7)$", "$\\psi(I_{\\omega+1})$", "$\\psi(I_{\\omega+2})$", "$\\psi(I_{\\omega+3})$", "$\\psi(I_{\\omega+4})$", "$\\psi(I_{\\omega 2})$", "$\\psi(I_{\\omega 3})$", "$\\psi(I_{\\omega 4})$", "$\\psi(I_{\\omega 5})$", "$\\psi(I_{\\Omega})$", "$\\psi(I_{\\psi_I(0)})$", "$\\psi(I_{\\psi(I)})$", "$\\psi(I_{\\psi(I^2)})$", "$\\psi(I_{\\psi(I^3)})$", "$\\psi(I_{\\psi(I^4)})$", "$\\psi(I_{I})$", "$\\psi(I_{I_{I}})$", "$\\psi(I_{I_{I_{I}}})$", "$\\psi(I_{I_{I_{I_{I}}}})$", "$\\psi(I_{I_{I_{I_{I_{I}}}}})$", "$\\psi(\\psi_{I(1,0)}(0))$", "$\\psi(\\psi_{I(1,0)}(1))$", "$\\psi(\\psi_{I(1,0)}(2))$", "$\\psi(\\psi_{I(1,0)}(3))$", "$\\psi(\\psi_{I(1,0)}(4))$", "$\\psi(\\psi_{I(1,0)}(5))$", "$\\psi(I(1,0))$", "$\\psi(I(1,0)+1)$", "$\\psi(I(1,0)+2)$", "$\\psi(I(1,0)+3)$", "$\\psi(I(1,0)+4)$", "$\\psi(I(1,0)+5)$", "$\\psi(I(1,0)2)$", "$\\psi(I(1,0)3)$", "$\\psi(I(1,0)4)$", "$\\psi(I(1,0)5)$", "$\\psi(I(1,0)^2)$", "$\\psi(I(1,0)^2 2)$", "$\\psi(I(1,0)^2 3)$", "$\\psi(I(1,0)^2 4)$", "$\\psi(I(1,0)^2 5)$", "$\\psi(I(1,0)^3)$", "$\\psi(I(1,0)^4)$", "$\\psi(I(1,0)^5)$", "$\\psi(I(1,0)^6)$", "$\\psi(I(1,0)^I)$", "$\\psi(I(1,0)^{I_2})$", "$\\psi(I(1,0)^{I_3})$", "$\\psi(I(1,0)^{I_4})$", "$\\psi(I(1,0)^{I_5})$", "$\\psi(I(1,0)^{I_I})$", "$\\psi(I(1,0)^{I_{I_{I}}})$", "$\\psi(I(1,0)^{I_{I_{I_{I}}}})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(0)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(1)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(2)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(3)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(4)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(5)})$", "$\\psi(I(1,0)^{I(1,0)})$", "$\\psi(I(1,0)^{I(1,0)^{I(1,0)}})$", "$\\psi(I(1,0)^{I(1,0)^{I(1,0)^{I(1,0)}}})$", "$\\psi(I(1,0)^{I(1,0)^{I(1,0)^{I(1,0)^{I(1,0)}}}})$", "$\\psi(I(1,1))$", "$\\psi(I(1,2))$", "$\\psi(I(1,3))$", "$\\psi(I(1,4))$", "$\\psi(I(1,5))$", "$\\psi(I(1,\\omega+1))$", "$\\psi(I(1,\\Omega))$", "$\\psi(I(1,\\psi_{I(1,0)}(0)))$", "$\\psi(I(1,\\psi(I(1,0))))$", "$\\psi(I(2,0))$", "$\\psi(I(2,1))$", "$\\psi(I(2,2))$", "$\\psi(I(2,3))$", "$\\psi(I(2,4))$", "$\\psi(I(2,5))$", "$\\psi(I(3,0))$", "$\\psi(I(4,0))$", "$\\psi(I(5,0))$", "$\\psi(I(6,0))$", "$\\psi(M^M)$", "$\\psi(M^M+1)$", "$\\psi(M^M+2)$", "$\\psi(M^M+3)$", "$\\psi(M^M+4)$", "$\\psi(M^M+5)$", "$\\psi(M^M+\\omega)$", "$\\psi(M^M+\\Omega)$", "$\\psi(M^M+\\psi_I(0))$", "$\\psi(M^M+\\psi_I(1))$", "$\\psi(M^M+\\psi_I(2))$", "$\\psi(M^M+\\psi_I(3))$", "$\\psi(M^M+M)$", "$\\psi(M^M+M^2)$", "$\\psi(M^M+M^3)$", "$\\psi(M^M+M^4)$", "$\\psi(M^M+M^5)$", "$\\psi(M^M2)$", "$\\psi(M^M3)$", "$\\psi(M^M4)$", "$\\psi(M^M5)$", "$\\psi(M^M6)$", "$\\psi(M^{M+1})$", "$\\psi(M^{M+2})$", "$\\psi(M^{M+3})$", "$\\psi(M^{M+4})$", "$\\psi(M^{M+5})$", "$\\psi(M^{M2})$", "$\\psi(M^{M3})$", "$\\psi(M^{M4})$", "$\\psi(M^{M5})$", "$\\psi(M^{M^2})$", "$\\psi(M^{M^2}+1)$", "$\\psi(M^{M^2}+2)$", "$\\psi(M^{M^2}+3)$", "$\\psi(M^{M^2}+4)$", "$\\psi(M^{M^2}+5)$", "$\\psi(M^{M^2}+\\omega)$", "$\\psi(M^{M^2}+\\Omega)$", "$\\psi(M^{M^2}+M)$", "$\\psi(M^{M^2}+M^2)$", "$\\psi(M^{M^2}+M^3)$", "$\\psi(M^{M^2}+M^4)$", "$\\psi(M^{M^2}+M^5)$", "$\\psi(M^{M^2}+M^M)$", "$\\psi(M^{M^2}2)$", "$\\psi(M^{M^2}3)$", "$\\psi(M^{M^2}4)$", "$\\psi(M^{M^2}5)$", "$\\psi(M^{M^2+1})$", "$\\psi(M^{M^2+2})$", "$\\psi(M^{M^2+4})$", "$\\psi(M^{M^2+5})$", "$\\psi(M^{M^3})$", "$\\psi(M^{M^4})$", "$\\psi(M^{M^5})$", "$\\psi(M^{M^6})$", "$\\psi(M^{M^M})$", "$\\psi(M^{M^{M^{M}}})$", "$\\psi(M^{M^{M^{M^{M}}}})$", "$\\psi(M^{M^{M^{M^{M^{M}}}}})$", "$\\psi(M^{M^{M^{M^{M^{M^{M}}}}}})$", "$\\psi(M^{M^{M^{M^{M^{M^{M^{M}}}}}}})$", "$\\psi(\\psi_{M_2}(0))$", "$\\psi(\\psi_{M_2}(1))$", "$\\psi(\\psi_{M_2}(2))$", "$\\psi(\\psi_{M_2}(3))$", "$\\psi(\\psi_{M_2}(4))$", "$\\psi(\\psi_{M_2}(5))$", "$\\psi(M_2)$", "$\\psi(M_2+1)$", "$\\psi(M_2+2)$", "$\\psi(M_2+3)$", "$\\psi(M_2+4)$", "$\\psi(M_2+5)$", "$\\psi(M_2+\\omega)$", "$\\psi(M_2+\\Omega)$", "$\\psi(M_2+M)$", "$\\psi(M_2 2)$", "$\\psi(M_2 3)$", "$\\psi(M_2 4)$", "$\\psi(M_2 5)$", "$\\psi(M_2 6)$", "$\\psi(M_2^2)$", "$\\psi(M_2^2+1)$", "$\\psi(M_2^2+2)$", "$\\psi(M_2^2+3)$", "$\\psi(M_2^2+4)$", "$\\psi(M_2^2+5)$", "$\\psi(M_2^2 2)$", "$\\psi(M_2^2 3)$", "$\\psi(M_2^2 4)$", "$\\psi(M_2^2 5)$", "$\\psi(M_2^3)$", "$\\psi(M_2^4)$", "$\\psi(M_2^5)$", "$\\psi(M_2^6)$", "$\\psi(M_2^M)$", "$\\psi(M_2^{M_2})$", "$\\psi(M_2^{M_2^{M_2}})$", "$\\psi(M_2^{M_2^{M_2^{M_2}}})$", "$\\psi(M_2^{M_2^{M_2^{M_2^{M_2}}}})$", "$\\psi(M_2^{M_2^{M_2^{M_2^{M_2^{M_2}}}}})$", "$\\psi(M_2^{M_2^{M_2^{M_2^{M_2^{M_2^{M_2}}}}}})$", "$\\psi(M_3)$", "$\\psi(M_4)$", "$\\psi(M_5)$", "$\\psi(M_6)$", "$\\psi(M_7)$", "$\\psi(M_8)$", "$\\psi(M_9)$", "$\\psi(M_{\\omega+1})$", "$\\psi(M_{\\omega+2})$", "$\\psi(M_{\\omega+3})$", "$\\psi(M_{\\omega+4})$", "$\\psi(M_{\\omega+5})$", "$\\psi(M_\\Omega)$", "$\\psi(M_{M})$", "$\\psi(M_{M_{2}})$", "$\\psi(M_{M_{3}})$", "$\\psi(M_{M_{4}})$", "$\\psi(M_{M_{5}})$", "$\\psi(M_{M_{M}})$", "$\\psi(M_{M_{M_{M}}})$", "$\\psi(M_{M_{M_{M_{M}}}})$", "$\\psi(M_{M_{M_{M_{M_{M}}}}})$", "$\\psi(M_{M_{M_{M_{M_{M_{M}}}}}})$", "$\\psi(M_{M_{M_{M_{M_{M_{M_{M}}}}}}})$", "$\\psi(\\psi_{M(1,0)}(0))$", "$\\psi(\\psi_{M(1,0)}(1))$", "$\\psi(\\psi_{M(1,0)}(2))$", "$\\psi(\\psi_{M(1,0)}(3))$", "$\\psi(\\psi_{M(1,0)}(4))$", "$\\psi(\\psi_{M(1,0)}(5))$", "$\\psi(\\psi_{M(1,0)}(\\omega))$", "$\\psi(\\psi_{M(1,0)}(\\omega+1))$", "$\\psi(\\psi_{M(1,0)}(\\omega+2))$", "$\\psi(\\psi_{M(1,0)}(\\omega+3))$", "$\\psi(\\psi_{M(1,0)}(\\omega+4))$", "$\\psi(\\psi_{M(1,0)}(\\omega+5))$", "$\\psi(\\psi_{M(1,0)}(\\Omega))$", "$\\psi(\\psi_{M(1,0)}(M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+1))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+\\omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+\\Omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)6))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)\\omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^6))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^\\omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^\\Omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^2}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^3}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^4}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^5}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}}))$", "$\\psi(M(1,0))$", "$\\psi(M(1,0)+1)$", "$\\psi(M(1,0)+2)$", "$\\psi(M(1,0)+3)$", "$\\psi(M(1,0)+4)$", "$\\psi(M(1,0)+5)$", "$\\psi(M(1,0)2)$", "$\\psi(M(1,0)3)$", "$\\psi(M(1,0)4)$", "$\\psi(M(1,0)5)$", "$\\psi(M(1,0)^2)$", "$\\psi(M(1,0)^3)$", "$\\psi(M(1,0)^4)$", "$\\psi(M(1,0)^5)$", "$\\psi(M(1,0)^6)$", "$\\psi(M(1,0)^{M(1,0)})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}}})$", "$\\psi(M(1,1))$", "$\\psi(M(1,2))$", "$\\psi(M(1,3))$", "$\\psi(M(1,4))$", "$\\psi(M(1,5))$", "$\\psi(M(1,6))$", "$\\psi(M(1,\\omega+1))$", "$\\psi(M(1,\\Omega))$", "$\\psi(M(1,M))$", "$\\psi(M(1,M(1,0)))$", "$\\psi(M(1,M(1,M(1,0))))$", "$\\psi(M(1,M(1,M(1,M(1,0)))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,0))))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,M(1,0)))))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,M(1,M(1,0))))))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,M(1,M(1,M(1,0)))))))))$", "$\\psi(M(2,0))$", "$\\psi(M(2,1))$", "$\\psi(M(2,2))$", "$\\psi(M(2,3))$", "$\\psi(M(2,4))$", "$\\psi(M(2,5))$", "$\\psi(M(2,\\omega))$", "$\\psi(M(2,\\Omega))$", "$\\psi(M(2,M))$", "$\\psi(M(2,M(1,0)))$", "$\\psi(M(2,M(1,M(1,0))))$", "$\\psi(M(2,M(1,M(1,M(1,0)))))$", "$\\psi(M(2,M(1,M(1,M(1,M(1,0))))))$", "$\\psi(M(2,M(2,0)))$", "$\\psi(M(2,M(2,M(2,0))))$", "$\\psi(M(2,M(2,M(2,M(2,0)))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,0))))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,M(2,0)))))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,M(2,M(2,0))))))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,M(2,M(2,M(2,0)))))))))$", "$\\psi(M(3,0))$", "$\\psi(M(4,0))$", "$\\psi(M(5,0))$", "$\\psi(M(6,0))$", "$\\psi(M(7,0))$", "$\\psi(M(M,0))$", "$\\psi(M(M(1,0),0))$", "$\\psi(M(M(M(1,0),0),0))$", "$\\psi(M(M(M(M(1,0),0),0),0))$", "$\\psi(M(M(M(M(M(1,0),0),0),0),0))$", "$\\psi(M(M(M(M(M(M(1,0),0),0),0),0),0))$", "$\\psi(M(M(M(M(M(M(M(1,0),0),0),0),0),0),0))$", "$\\psi(M(M(M(M(M(M(M(M(1,0),0),0),0),0),0),0),0))$", "$\\psi(M(1;0)^{M(1;0)})$", "$\\psi(M(1;0)^{M(1;0)}+1)$", "$\\psi(M(1;0)^{M(1;0)}+2)$", "$\\psi(M(1;0)^{M(1;0)}+3)$", "$\\psi(M(1;0)^{M(1;0)}+4)$", "$\\psi(M(1;0)^{M(1;0)}+5)$", "$\\psi(M(1;0)^{M(1;0)}+M)$", "$\\psi(M(1;0)^{M(1;0)}+M_2)$", "$\\psi(M(1;0)^{M(1;0)}+M_3)$", "$\\psi(M(1;0)^{M(1;0)}+M_4)$", "$\\psi(M(1;0)^{M(1;0)}+M_5)$", "$\\psi(M(1;0)^{M(1;0)}+M_6)$", "$\\psi(M(1;0)^{M(1;0)}+M(1,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,1))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,2))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,3))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,4))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,5))$", "$\\psi(M(1;0)^{M(1;0)}+M(2,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(3,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(4,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(5,0))$", "$\\psi(M(1;0)^{M(1;0)}2)$", "$\\psi(M(1;0)^{M(1;0)}3)$", "$\\psi(M(1;0)^{M(1;0)}4)$", "$\\psi(M(1;0)^{M(1;0)}5)$", "$\\psi(M(1;0)^{M(1;0)}\\omega)$", "$\\psi(M(1;0)^{M(1;0)}M)$", "$\\psi(M(1;0)^{M(1;0)}M(1,0))$", "$\\psi(M(1;0)^{M(1;0)}M(2,0))$", "$\\psi(M(1;0)^{M(1;0)}M(3,0))$", "$\\psi(M(1;0)^{M(1;0)}M(4,0))$", "$\\psi(M(1;0)^{M(1;0)}M(5,0))$", "$\\psi(M(1;0)^{M(1;0)+1})$", "$\\psi(M(1;0)^{M(1;0)+2})$", "$\\psi(M(1;0)^{M(1;0)+3})$", "$\\psi(M(1;0)^{M(1;0)+4})$", "$\\psi(M(1;0)^{M(1;0)+5})$", "$\\psi(M(1;0)^{M(1;0)2})$", "$\\psi(M(1;0)^{M(1;0)3})$", "$\\psi(M(1;0)^{M(1;0)4})$", "$\\psi(M(1;0)^{M(1;0)5})$", "$\\psi(M(1;0)^{M(1;0)^{2}})$", "$\\psi(M(1;0)^{M(1;0)^{3}})$", "$\\psi(M(1;0)^{M(1;0)^{4}})$", "$\\psi(M(1;0)^{M(1;0)^{5}})$", "$\\psi(M(1;0)^{M(1;0)^{M}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1,0)}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}}}}})$", "$\\psi(M(1;1))$", "$\\psi(M(1;1)2)$", "$\\psi(M(1;1)3)$", "$\\psi(M(1;1)4)$", "$\\psi(M(1;1)5)$", "$\\psi(M(1;1)^2)$", "$\\psi(M(1;1)^3)$", "$\\psi(M(1;1)^4)$", "$\\psi(M(1;1)^5)$", "$\\psi(M(1;1))$", "$\\psi(M(1;2))$", "$\\psi(M(1;3))$", "$\\psi(M(1;4))$", "$\\psi(M(1;5))$", "$\\psi(M(1;\\omega))$", "$\\psi(M(1;M))$", "$\\psi(M(1;M(1;0)))$", "$\\psi(M(1;M(1;M(1;0))))$", "$\\psi(M(1;M(1;M(1;M(1;0)))))$", "$\\psi(M(1;M(1;M(1;M(1;M(1;0))))))$", "$\\psi(M(1;1,0)$", etc. ] Tar(n) = f_{\underbrace{C(C(\cdots(C(C(\Omega_{n}2,0),0),\cdots ),0)}_{n\text{ C's}}}(n) Example of an LNGI/Sequence LaTeX codes: Ultimate List of Large Numbers and Ordinals: ψ(ψ_Ι(0)) to ψ(ε_{Ξ+1}) and beyond up to ψ(C;C;C;....;C) = Church-Kleene Ordinal (e.g. \begin{array}{l}\psi(\psi_I(0)) \\ \psi(\psi_I(0)+1) \\ \psi(\psi_I(0)+\omega) \\ \psi(\psi_I(0)+\Omega) \\ \psi(\psi_I(0)2) \\ \psi(\psi_I(0)\omega) \\ \psi(\psi_I(0)^2) \\ \psi(\psi_I(0)^\omega) \\ \psi(\psi_I(0)^\Omega) \\ \psi(\psi_I(0)^{\psi_I(0)}) \\ \psi(\psi_I(0)^{\psi_I(0)^{\psi_I(0)}}) \\ \psi(\psi_I(1)) \\ \psi(\psi_I(\omega)) \\ \psi(\psi_I(\Omega)) \\ \psi(\psi_I(\psi_I(0))) \\ \psi(I) \\ \psi(I+1) \\ \psi(I+\omega) \\ \psi(I+\Omega) \\ \psi(I+\psi_I(0)) \\ \psi(I2) \\ \psi(I\omega) \\ \psi(I^2) \\ \psi(I^\omega) \\ \psi(I^\Omega) \\ \psi(I^I) \\ \psi(I^{I^I}) \\ \psi(I_2) \\ \psi(I_2+1) \\ \psi(I_2+\omega) \\ \psi(I_2 2) \\ \psi(I_2^2) \\ \psi(I_2^{I_2}) \\ \psi(I_3) \\ \psi(I_{\omega}) \\ \psi(I_{\Omega}) \\ \psi(I_I) \\ \psi(I(1,0)) \\ \psi(I(1,0)+1) \\ \psi(I(1,0)2) \\ \psi(I(1,0)^2) \\ \psi(I(1,0)^{I(1,0)}) \\ \psi(I(1,1)) \\ \psi(I(2,0)) \\ \psi(I(I,0)) \\ \psi(M) \\ \psi(M+1) \\ \psi(M+\omega) \\ \psi(M+\Omega) \\ \psi(M+I) \\ \psi(M2) \\ \psi(M\omega) \\ \psi(M^2) \\ \psi(M^\omega) \\ \psi(M^\Omega) \\ \psi(M^I) \\ \psi(M^M) \\ \psi(M^{M^M}) \\ \psi(M_2) \\ \psi(M_2+1) \\ \psi(M_2+\omega) \\ \psi(M_2 2) \\ \psi(M_2^2) \\ \psi(M_2^{M_2}) \\ \psi(M_3) \\ \psi(M_{\omega}) \\ \psi(M_{\Omega}) \\ \psi(M_M) \\ \psi(M(1,0)) \\ \psi(M(1,0)+1) \\ \psi(M(1,0)2) \\ \psi(M(1,0)^2) \\ \psi(M(1,0)^{M(1,0)}) \\ \psi(M(1,1)) \\ \psi(M(2,0)) \\ \psi(M(M,0)) \\ \psi(M(M(1,0),0)) \\ \psi(M) \end{array}) Tier 2 The document starts at millillion, to give the audience an overview. Tier 1 Index Tier 2 Index Name Abbreviation 1,000 1 millilllion Mi 1,001 milli-unilllion Mi-U 1,002 milli-duilllion Mi-D 1,003 milli-trilllion Mi-T 1,004 milli-quadrilllion Mi-Qa 1,010 milli-decilllion Mi-De 1,100 milli-centilllion Mi-Ce 2,000 dumillilllion DMi 2,001 dumilli-unilllion DMi-U 3,000 trimillilllion TMi 4,000 quadrimillilllion QaMi 10,000 decimillilllion DeMi 11,000 undecimillilllion UDeMi 20,000 vigintimillilllion VgMi 100,000 centimillilllion CeMi Index Name (Myrillions) Abbreviation 10,000 myrillion My 11,000 myrimillillion My-Mi 20,000 dumyrillion DMy 100,000 decimyrillion DeMy 1,000,000 2 micrilllion Mc 1,000,001 micri-unilllion Mc-U 1,001,000 micri-millilllion Mc-Mi 1,001,001 micri-milli-unilllion Mc-Mi-U 2,000,000 dumicrilllion DMc 10,000,000 decimicrillion DeMc 1,000,000,000 3 nanilllion Na 1,000,000,001 nani-unillion Na-U 2,000,000,000 dunanillion DNa 1,000,000,000,000 4 picillion Pi / Pc 1,000,000,000,000,000 5 femtillion Fem My extension starts at an attillion. So, get ready! (also includes the old version, which is always capitalized) Tier 2 Index Name Abbreviation Old version 5 femtilllion Fem FM 6 attilllion At AT 7 zeptilllion Zep ZP 8 yoctilllion Yo YT 9 rontillion Rt RT 10 quectillion Qu QE 11 mecilllion Me(c) ME 12 duecilllion Du(c) DE 13 trecilllion Tre(c) TE 14 tetrecilllion Te(c) TeE 15 pentecilllion Pe(c) PE 16 hexecilllion He(c) HE 17 heptecilllion Hp(c) HeE 18 octecilllion Ot(c) OC 19 ennecilllion En(c) EC 20 icosilllion Is(o) IS 21 meicosilllion MeIs MS 22 dueicosilllion DuIs DS 23 trioicosilllion TreIs TS 24 tetreicosilllion TeIs TeS 25 penteicosilllion PeIs PS 26 hexeicosilllion HeIs HS 27 hepteicosilllion HpIs HeS 28 octeicosilllion OtIs OS 29 enneicosilllion EnIs ES 30 triacontilllion TrC / TCn TN 31 metriacontilllion MeTrC MTN 32 duetriacontilllion DuTrC DTN 1 Using BIPM’s 2022 proposal. Large quantitative prefixes are ronna and quetta. (formerly quecca) You might see significant differences as I go further from traditional -illions. The abbreviations of unit roots have been changed drastically to avoid ambiguous abbreviations easier. The E letter from 10-17th tier 2 roots has been changed into C. The icosi- root now always stands for IC, regardless of the following tier 2 units. Same goes to the following “tens” roots: triaconta-, tetraconta-, etc. Let’s speed up the work. Tier 2 Index Name Abbreviation Old version 10 quectilllion Qc(t) VE 20 icosilllion Is(o) IS 30 triacontilllion TrC TN 40 tetracontilllion TeC TeC 50 pentacontilllion PeC / PCo PC 60 hexacontilllion HeC / HCt HC 70 heptacontilllion HpC HeC 80 octacontilllion OtC / OCn OC 90 enneacontilllion EnC / ECo EC 100 hectilllion Hec HT 101 mehectilllion MeHec MHT 102 duehectilllion DuHec DHT (limit in NG+3 standalone) 110 quectehectilllion QtHc QEHT 111 mecehectilllion MecHc MEHT 112 duecehectilllion DucHc MEHT 120 icosehectilllion IsHc ISHT 200 dohectilllion DHc DT 210 quectedohectilllion QtDHc QEDT 300 triahectilllion TrH / THt TT 400 tetrahectilllion TeH TeT 500 pentahectilllion PeH / PHc PT 600 hexahectilllion HeH / HHe HT 700 heptahectilllion HpH / HpH HeT 800 octahectilllion OtH / OHt OT 900 enneahectilllion EnH / EHc ET 990 enneconteenneahectilllion EnCEnH ECET 999 enneenneconteenneahectilllion = 10^{3 \times 10^{2,997}+3} (not always 10^3000) EnEnCEnH EECET That’s the Standard limit in NG+3R formatting. However, it’s not over yet. Now, let’s go to bigger, hyperdimensional names. Tier 1 (Maximus) Before I move on to Tier 3, let’s abbreviate some maximus- numbers. Exponent Maximus Index Name Abbreviation 1,000 0 maximusthousand MXS-K 1,000,000 1 maximusmillion MXS-M 1,000,000,000 2 maximusbillion MXS-B 1,000,000,000,000 3 maximustrillion MXS-T By the way, 10^x = MXS-x. Tier 3 Do you know that in my Standard conversion, it stops at nanepetillion? There are no tier 3 changes, except terillions. Tier 2 Index Tier 3 Name Abbreviation 1,000 1 killillion Kl 1,000+ 1+ killo-millillion Kl-Mi 1,000* dukillillion DKl 1,001 killa-millillion Kla’Mi 1,002 killa-micrillion Kla’Mc 1,010 killa-quectillion Kla’Qu 1,100 killa-hectillion Kla’Hc 2,000 1* micrekillillion duekillillion McKl DuKl 3,000 nanekillillion triokillillion NaKl TrKl 10,000 quectekillillion QuKl 100,000 hectekillillion HecKl 1,000,000 2 megillion Mg 1,000,001 2+ mega-millillion Mga’Mi 1,001,000 mega-killillion Mga’Ki 2,000,000 2* micremegillion duemegillion McMg DuMg 1,000,000,000 3 gigillion Gi(g) 1,000,000,000,000 4 terillion Tr / Ter 1,000,000,000,000,000 5 petillion Pt 3,000,000,000,000,000 5* nanepetillion triopetillion TrPt For now on, I have to make up new abbreviations to go up to the limit. Tier 3 Name Abbreviation 6 exillion Ex 7 zettillion Zt 8 yottillion Yt 9 ronnillion Rn 10 quettillion Qt 11 hendillion Hn / He 12 dokillion Dok 13 tradakillion TrD 14 tedakillion TeD 15 pedakillion PeD 16 exdakillion ExD 17 zedakillion ZeD 18 yodakillion YoD 19 nedakillion NeD 20 ikillion Ik / IK 21 ikenillion IkeN 22 icodillion IcoD 23 ictrillion IctR 24 icterillion IctE 25 ikpetillion IkpT 26 ikexillion IkeX 27 iczetillion IczE 28 ikyotillion IkyO 29 icronillion IcrN 30 trakillion Trk / TrK 31 trakenillion TkeN / TrkeN 32 tracodillion TcoD / TrkoD 33 tractrillion TctR / TrktR 40 tekillion Tek / TeK 41 tekenillion TekeN 50 pekillion Pek / PeK 51 pekenillion PkeN / PekeN 60 exakillion Exk / ExK 61 exakenillion EkeN / ExkeN 70 zakillion Zak / ZaK / Za 71 zakenillion ZkeN / ZakeN 80 yokillion Yok / YoK 81 yokenillion YkeN / YokeN 90 nekillion Nek / NeK 91 nekenillion NkeN / NekeN 99 necronillion NcrN / NecrN 100 hotillion HoT 101 hotenillion HoEN 102 hotodillion HoOD 103 hotrillion HoTR 104 hoterillion HoTE 105 hopetillion HoPT 106 hotectillion HoEC 107 hozetillion HoZT 108 hoyotillion HoYT 109 horonillion HoRN 110 hoquettillion HoQE 111 hotendillion HoEn 112 hodokillion HoDoK 113 hotradakillion HoTrD 114 hotedakillion HoTeD 115 hopedakillion HoPeD 116 hoexdakillion HoExD 117 hozedakillion HoZeD 118 hoyodakillion HoYoD 119 honedakillion HoNeD 120 hotikillion HoIK 121 hotikenillion HoIkeN 130 hotrakillion HoTrK 160 hotexakillion HoTeK 200 dotillion DoT 300 totillion ToT 400 trotillion TrT 500 potillion PoT 600 exotillion ExT 700 zotillion ZoT 800 yootillion YoT 900 notillion NoT 990 nonekillion NoNK 999 nonecronillion NoNcrN Time to blast off to the astronomical step! Tier 4 Now, we are approaching outer space. We will now see the hypercelestial objects... The machine seems to collect the celestial dust and create more powerful abbreviations. After months of break on this document, I am finally introducing Tier 4! You know the drill, KaL = kalillion, MeJ = mejillion, GiJ = Gijillion, etc. Remove the last letter but except coming before eN, oD, and tR. Tier 3 Name Abbreviation 1,000 kalillion KaL 1,001 kalenillion KaLeN 1,002 kalodillion KaLoD 1,003 kaltrillion KaLtR 1,004 kalterillion KaTer 1,005 kalpetillion KaPt 1,006 kalexillion KaEx 1,007 kalzettillion KaZt 1,008 kalyottillion KaYt 1,009 kalronnillion KaRn 1,010 kalquettillion KaQu 1,011 kalhendillion KaHn 1,020 kalikillion KaIk 1,030 kaltrakillion KaTrk 1,100 kalhotillion KaHoT 1,999 kalnonecronillion KaNoNcrN 2,000 dalillion DaL 2,001 dalenillion DaLeN 2,010 daldakillion DaDk 2,100 dalhotillion DaHoT 3,000 talillion TaL 4,000 tralillion TraL 5,000 palillion PaL 6,000 exalillion ExaL 7,000 zalillion ZaL 8,000 yalillion YaL 9,000 nalillion NaL 10,000 dakalillion DKaL 11,000 hendalillion HNaL 20,000 ikalillion IKaL 100,000 hotalillion HoTaL 1,000,000 mejillion MeJ 1,000,001 mejenillion MeJeN 1,001,000 mejkalillion MeKaL 1,010,000 mejdakalillion MeDKaL 2,000,000 dejillion DeJ 1,000,000,000 gijillion GiJ 2,000,000,000 dijillion DiJ Tier 4 Name Abbreviation* 4 astillion AsT 5 lunillion LuN 6 fermillion FrM 7 jovillion JoV 8 solillion SoL 9 betillion BeT** 10 glocillion GlO 11 gaxillion GaX 12 supillion SuP 13 versillion VrS 14 multillion MlT After gijillion, add ` instead of removing the last letter in a process. ** When tier 3 roots come before betillion, replace e with ee to follow Douglas’ fixes. [Extended Tier 4 coming soon] although barrillion = BrRl, barrekalillion = BrKaL [Tier 5 coming soon] although hepillion = HepA, geopillion = GepA [Tier 6 coming soon] although hapaxillion = HPx, tetrakisillion = TeKs [Tier 7 coming soon] although redillion = Rd, icoise-resepi-decakisillion = IseRp-DcKs [[ INTRODUCTION ]] From early history, humans thought of an extensible naming system for grouping large numbers. That is, in English, we group with millions, billions, and trillions. Around the 20th century, mathematicians proposed new illions that haven't been seen from the real world before. In the early 2000s, Bowers decided to start his extensive list of illions, including numbers that reach beyond the realistic range. Sbiis, on the other hand, released the extensible system for Bowers’ -illions, with all naming gaps filled out. Sbiis’ version Out of curiosity, Douglas pointed out there was a small mistake that the petillion is ambiguous from his list. He fixed the problem by putting another e for tier 3 multipliers to -et root. He also published the LNGI page for Sbiis’ illions which is shown below. Numbers Getting Bigger - illions version Nearly a year after the reformation began, DeepLineMadom fixed the ambiguity problem for yocillion too! The earlier confusing -illions were fixed by adding “i” in-between several root connections. With problems fixed, we could name numbers up to the limit of Bowers’ illions, which is the supremum of all tier 0 - 4 roots being used. In this page, I am going to take Sbiis’ and Douglas’ spirits and continue the journey of never-ending illions list. Get ready, let’s reach out of the ethereal illions! [EXTENSIONS] Rise out of Bower’s illions, there are joints to different possibilities! One is spreading out to a tetrational range which metaness can’t keep up! In early 2020s, Nirvana (now CompactStar) decided to extend, up to the ridiculous limit of the Tier 6 system! Regardless of only milestones, some existing names are gathered for amalgamates… CompactStar’s -illions Not only that, there’s many illions going after multillion, like metillion, pyrillion, metalillion, and so infinitum. Tier 5: hepillion / febrillion / keltillion / wektillion Tier 6: hapaxillion Tier 7: redillion Tier 8: erkillion Tier 9: izzillion Tier 10: meakoowillion However, I will only fill out his system and some more, even going beyond Tier 6! I will also fix his Tier 5 system too because of the predefined connections too. (geopahepillion is already precoined, by [Tier 2] connections between geopillion and hepillion) (Tier 2 additive: geop + (a) + hep -> geopahepillion) With connections ready, let’s aim at the naming gaps. [ILLION FUNCTION] Unfortunately, the Extensible Illion System is ill-defined. I have to define a new function by myself: Ha = Ha#1 Ha#0 = 1,000*a b > 0: Ha#b = H(1,000^a)#(b-1) Examples: H34 = H34#1 = H(1,000^34)#0 = 1,0001,000^34 = 10^105 = 1 quattourtrigintillion H2,013#2 = H(1,000^2,013)#1 = 1,000^(1,000^2,013+1) = 10^(310^6,039+3) = 1 micrekillatrecillion Illion Function can also be expressed in Hyper-E: Ha#b = 1,000E(1,000)a#b And can be estimated approximately: 1,000^^(b+1) > Ha#b > 1,000^^b (for a < 333) Yes, this can be tweaked into a variant of Sbiis’ Extensible-E Notation. All it needs is a simple rule. I denote E as an ExE base. Hn -> 1,000*E(1,000)n [[ LIST OF ILLIONS ]] [Tier 4] - HYPERCELESTIAL :[ 10^10^10^10^45 - 10^10^10^10^3000 ]: :[ IMPROVED BY DEEPLINEMADOM, FIXED BY AAREX ]: Hang on… How I start at metillion? Guess what, the naming gaps were already filled. I was going meta for nonstandard illions whatsoever. But, one ubiquitous issue. If I take met and put the first letter off, it would result the same as I do to bet. They are splitted into 2 identical "et." (taking the first letter out: bet -> et, met -> et) But did I stump on that? NO. I want to do a similar resolution as what Douglas resolved. He decided to turn et and eet when the first letter (b) is removed. This is consistent as you are adding Tier 3 multiplicatives to Tier 4. (1st: Taking the first letter out: bet -> et, 2nd: Douglas’ resolution, adding another e: et -> eet, 3rd: Finally, add Tier 3 roots as multiplicative: hot + eet -> hoteet) This doesn’t work for other tier multiplicatives and all additives: (Tier 1 multiplicative: vigint + (i) + bet -> vigintibet) (Tier 2 multiplicative: duec + (e) + bet -> duecebet) (Tier 1 additive: bet + (o) + zept -> betozept) (Tier 2 additive: bet + (a) + xenn -> betaxenn) (Tier 3 additive: bet + mej -> betmej) I eradicate met into ett for Tier 3 multiplicatives to met. In that case, take out the first letter and add another t after e. (d + met - (m) + (t) -> dettillion) (tr + met - (m) + (t) -> trettillion) (t + met - (m) + (t) -> tettillion) (sequence: metillion, dettillion, trettillion, tettillion, …) Miserly, one collides with the existing zettillion… (z + met - (m) + (t) -> zettillion: collision detected) So the resolved variant for zettillion has to be degenerated to zeettillion. Just add another e, like Douglas! (z + (e) + met - (m) + (t) -> zeettillion) Now I can go hypercelestial and reach far out of natural cosmology! Let's blast at speeds of abstraction! Tier 3 Index Number Name Tier 3 Index Number Name 1027 H(110^27)#3 10^(310^(310^(310^27))+3) betillion 1045 H(110^45)#3 10^(310^(310^(310^45))+3) metillion 21027 H(210^27)#3 10^(310^(310^(610^27))+3) deetillion 21045 H(210^45)#3 10^(310^(310^(610^45))+3) dettillion 31027 H(310^27)#3 10^(310^(310^(910^27))+3) treetillion 31045 H(310^45)#3 10^(310^(310^(910^45))+3) trettillion 41027 H(410^27)#3 10^(310^(310^(1210^27))+3) teetillion 41045 H(410^45)#3 10^(310^(310^(1210^45))+3) tettillion 51027 H(510^27)#3 10^(310^(310^(1510^27))+3) peetillion 51045 H(510^45)#3 10^(310^(310^(1510^45))+3) pettillion 61027 H(610^27)#3 10^(310^(310^(1810^27))+3) exeetillion 61045 H(610^45)#3 10^(310^(310^(1810^45))+3) exettillion 71027 H(710^27)#3 10^(310^(310^(2110^27))+3) zeetillion 71045 H(710^45)#3 10^(310^(310^(2110^45))+3) zeettillion (resolved) 81027 H(810^27)#3 10^(310^(310^(2410^27))+3) yeetillion 81045 H(810^45)#3 10^(310^(310^(2410^45))+3) yettillion 91027 H(910^27)#3 10^(310^(310^(2710^27))+3) neetillion 91045 H(910^45)#3 10^(310^(310^(2710^45))+3) nettillion 101027 H(1010^27)#3 10^(310^(310^(3010^27))+3) dakeetillion 101045 H(1010^45)#3 10^(310^(310^(3010^45))+3) dakettillion 111027 H(1110^27)#3 10^(310^(310^(3310^27))+3) hendeetillion 111045 H(1110^45)#3 10^(310^(310^(3310^45))+3) hendettillion Welcome to the next level… The list of illions go wild out! o_0` Now eradicate the first letters and mutate it on higher roots! This goes legionically worse; Root 0s 10s Multiplied Root 0s 10s +0 gloc +0 -oc +1 kal gax +1 -al -ax +2 mej sup +2 -ej -up +3 gij vers +3 -ij -ers +4 ast mult +4 -ast -ult +5 lun met +5 -un -ett +6 ferm xev +6 -erm -ev +7 sol hyp +7 -ol -(oh)yp +8 jov omni(v) +8 -ov -omni(v) +9 bet out +9 -eet -ut [1] With omnillion / omnivillion consensus, having a (v) is optional. [2] (oh) is optional too. Onto the local scale of barrels and glocs (globular cluster); Repeat the process for placing a tier 3 multiplicative by removing the first letter. Besides with Tier 3 multiplicatives to Tier 4… Let's fold the Tier 4 roots by generalizations. It has been recently tweaked to use most of the tier 4 factors instead. Root Ones Tens 0 (use the base roots instead) 1 -al 2 -ej barr- 3 -ij gic- 4 -ast* asc-* 5 -un luc- 6 -erm ferc- 7 -ov joc- 8 -esol solc-* 9 -et bec- [1]* The 24th tier-4 root is actually barreast. “Barrast” is ambiguous in Fonster’s system. [2] “-esol” remains unchanged for pronunciation. [3] Like astillion, do not remove “a” from asc-. [4] soc- is renamed to solc- due to collision to joc- with tier 3 multiplicatives. This allows us to name illions up to exactly cetillion, at 100th Tier 4! Using Bower’s polytope naming with generalized Compact Star’s -illions, the highest dimension before polyhuton is named: poly nonacxenecet-nonacxenecesol-nonacxenecov- nonacxenecerm-nonacxenecun-nonacxenecast- nonacxenecij-nonacxenecej-nonacxenecal- nonacxenec- nonacxenolcet-nonacxenolcesol-nonacxenolcov-- ... ... ... nonacxenarr-nonacxenut-nonacxenomn- nonacxenohyp-nonacxenev-nonacxenett- nonacxenult-nonacxeners-nonacxenup- nonacxenax- nonacxenoc-nonacxeneet-nonacxenol- nonacxenov-nonacxenerm-nonacxenaun- nonacxenast-nonacxenij-nonacxenej- nonacxenal-nonacxenon (10300-1-dimensional polytope) Let's take a major step up from fictional cosmology, and observe the farthests. "Cet Em' Down, Jut no Buts!" ~ Proto-Alien from Type 9.7 Archverse "Lut the Hundreds up, delete the (T)s, and Gut the Ones and Tens on the right! That's we build the Huts of Hypercelestial Scales!" “And let the Celestial Entity fix the “t” conflict after hundreds!” This is extraordinary… If you didn't understand, check the root table below: Root Ones (+10) [1] (+tens) [1] Tens Hundreds [2] 0 gloc (use the base roots instead) 1 kal gax al cet 2 mej sup ej barr mejet2 3 gij vers ij gic git 4 ast mult east2 asc aset 5 lun met un luc lut 6 ferm xev erm ferc fert 7 jov hyp ov joc jovt 8 sol omn(iv) sol/esol1 solc1 solt2 9 bet out et bec bect2 Tier 3 Connection [3] Ones Ones (+10) Tens Hundreds 0 oc (use the base roots instead) 1 al ax ocet*1 2 ej up arr ejet 3 ij ers ic it 4 ast ult asc aset 5 un eet uc unt 6 erm ev erc ert 7 ov ohyp oc ovt*2 8 ol omni olc olt 9 et ut ec ect*2 [1] Use (+10) for 10th - 19th teens roots. Else, use (+tens). [2] Remove the final “t” if it follows with “ones” or “tens” roots. [3] Only if it precedes with Tier 3 root. [*1] Renamed due to pronunciation [*2] Renamed due to conflicts. Examples (201) mejekalillion (410) aseglocillion (621) ferbarralillion (300) gitillion (700) jovtillion Now we organize the huts of hundreds, here’s the list of all tier-4 -illions! Number Tier 3 Index Tier 4 Index Name H1#4 = 10^(310^(310^3,000))+3) 1,000 1 kalillion H(10^3,000+1)#2 = 10^(310^(310^3,000+3))+3) 1,000+ 1+ kala-killillion H(210^3,000)#2 = 10^(310^(6*10^3,000))+3) 1,000+ duekalillion / micrekalillion H1,001#3 = 10^(310^(310^3,003))+3) 1,001 kalenillion H1,002#3 = 10^(310^(310^3,006))+3) 1,002 kalodillion H1,003#3 1,003 kaltrillion H1,004#3 1,004 kalterillion 1,010 kaldakillion 1,011 kaltendillion 1,020 kalikillion 1,100 kalhotillion H2,000#3 2,000 1* dalillion 2,001 dalenillion 3,000 tralillion H10,000#3 10,000 dakalillion 11,000 hendalillion 20,000 ikalillion H100,000#3 100,000 hotalillion H2#4 = H1,000,000#3 = 10^(310^(310^3,000,000))+3) 1,000,000 2 mejillion 1,000,001 2+ mejenillion 1,001,000 mejkalillion 1,001,001 mejkalenillion 1,002,000 mejdalillion 2,000,000 2* dejillion 10,000,000 dakejillion 100,000,000 hotejillion H3#4 = 10^(310^(310^(3*10^9)))+3) 1,000,000,000 3 gijillion 1,001,000,000 3+ gijmejillion H4#4 = 10^(310^(310^(3*10^12)))+3) 1,000,000,000,000 4 astillion 1,001,000,000,000 4+ astgijillion H5#4 = 10^(310^(310^(3*10^15)))+3) 1,000,000,000,000,000 5 lunillion 1016 5* dakunillion H6#4 1018 6 fermillion 1019 6* dakermillion H7#4 1021 7 jovillion 1022 7* dakovillion H8#4 1022 8 solillion 1024 8* dakolillion H9#4 1025 9 betillion 1027 9* dakeetillion H10#4 1028 10 glocillion 1030 10* dakocillion H11#4 1031 11 gaxillion 1033 11 dakaxillion H12#4 1034 12 supillion 1036 12 dakupillion H13#4 1037 13 versillion 1039 13 dakersillion H14#4 1042 14 multillion 1043 14 dakultillion 9991042 14 nonecxenultillion H15#4 1045 15 metillion 1046 15* dakettillion H16#4 1048 16 xevillion 1049 16* dakevillion H17#4 1051 17 hypillion 1052 17* dakypillion H18#4 1054 18 omnillion 1055 18* dakomnillion H19#4 1057 19 outillion 1058 19* dakoutillion H20#4 1060 20 barrillion 1061 20* dakarrillion H21#4 1063 21 barralillion H22#4 1066 22 barrejillion H23#4 1069 23 barrijillion H24#4 1072 24 barrastillion H25#4 1075 25 barrunillion H26#4 1078 26 barrermillion H27#4 1081 27 barrovillion H28#4 1084 28 barresolillion H29#4 1087 29 barretillion H30#4 1090 30 gicillion 1091 30* dakicillion H31#4 1093 31 gicalillion H32#4 1096 32 gicejillion H33#4 10120 40 ascillion 10121 40* dakascillion H30#4 10150 50 lucillion 10151 50* dakucillion H60#4 10180 60 fercillion 10181 60* dakercillion H70#4 10210 70 jocillion H80#4 10240 80 solcillion H90#4 10270 90 becillion H99#4 10297 99 becetillion H100#4 10300 100 cetillion 10301 100* dakocetillion H101#4 10303 101 cekalillion H102#4 10306 102 cemejillion H110#4 10330 110 cegaxillion H120#4 10360 120 cebarrillion H121#4 10363 121 cebarralillion H130#4 10390 130 cegicillion H140#4 10420 140 ceascillion H200#4 10600 200 mejetillion 10601 200* dakejetillion H300#4 10900 300 gitillion H400#4 101,200 400 asetillion H500#4 101,500 500 lutillion H600#4 101,800 600 fertillion H700#4 102,100 700 jovtillion H800#4 102,400 800 soltillion H900#4 102,700 900 bectillion H990#4 102,970 990 becbecillion H999#4 102,997 999 becbecetillion Before we get extraordinary, let's peek into other proposals for extended Bowers' illions. [Other Systems] There's so many possibilities from obscure websites. They are even organized in a sheet. It is always fun, so why not explain in this page? There is another name for 15th tier-4 -illion which is familiar to most googologists: pyrillion. I think this is a combination of pyro and -illion, but let’s get over it. Intermediate googologists just remove “p” for tier 3 multiplicatives. This is what I did: “The second letter is y, so I use the hypillion method. Just add an o- at the left of pyr- root!” This coins dopyrillion, which is 2*1045th tier-3 -illion, and several others. But, is that it? Wrong. There are so many googologisms left to go. Here are the possibilities for tier 4. Tier 4 Index Possibility 1 Possibility 2 Possibility 3 (by MultiSoul) Possibility 4 15 pyrillion (met) metalillion 16 guntillion xenillion xenoxillion 17 kentrillion hypeillion hyperillion 18 onlillion (omni) omnivillion 19 paptrillion supeillion (not confused with supillion) outsillion 20 housillion dupeillion (not confused with dupillion) barrelillion idixillion 21 horsillion / housalillion barrelalillion 22 barrelejillion 23 barrelijillion 24 barrelestillion (barrelastillion is predefined as tier-3 -illion) 25 barrelunillion 26 barrelermillion 27 barrelovillion 28 barrelolillion 29 barreletillion 30 trongillion garrelillion gradixillion 40 batillion astarrelillion asixillion 50 handrillion lunarrelillion lenixillion 60 mastillion fermarrelillion ferixillion 70 gotillion jovarrelillion javixillion 80 kortillion solarrelillion solixillion 90 nenkillion (from other site) Possibility 5 betarrelillion betixillion 100 janillion horillion huutillion 200 mejanillion morillion muutillion 300 gijanillion guutillion 400 astanillion astuutillion 500 fermanillion luutillion 600 lunanillion fuutillion 700 jovanillion juutillion 800 solanillion suutillion 900 betanillion buutillion Sigh, here I go. Possibility 1 is easy to generalize, because you can use these words, previous tier 4 multiplicatives (*even to marillions), to make intermediate values. Root Ones Ones (+10) Ones (+20) Tens Hundreds Root (tier 3 multiplicatives / tier 4 additives) Ones Ones (+10) Tens Hundreds 0 gloc (use the base roots instead) 0 -oc (use the base roots instead) 1 kal gax hors (optional) jan 1 -al -ax -an 2 mej sup hous mejan 2 -ej -up -ous -ejan 3 gij vers trong gijan 3 -ij -ers -ong -ijan 4 ast mult bat astan 4 -(a/e)st -ult -at -astan 5 lun pyr handr lunan 5 -un -(o/e)pyr -andr -unan 6 ferm gunt mast ferman 6 -erm -unt -(o/e)mast (to avoid with -ast) -erman 7 jov kentr got jovan (to avoid with jar) 7 -ov -entr -(o/e)got (to avoid with -ot) -ovan 8 sol onl kort solan 8 -ol -eonl -ort -olan 9 bet paptr nenk betan 9 -eet / -et -aptr -enk -etan The limit for Possibility 1 is betanenketillion. I made up for Possibility 1a, for using tier 4 multiplicatives to marillions for “hundreds” roots. Root Hundreds Hundreds (tier 3 multiplicatives) 0 1 jan -an 2 mejan -ejan 3 gan -ogan 4 asan -asan 5 lan -olan 6 fan -ofan 7 jovan (to avoid with jar) -ovan 8 san -olan 9 ban -etan Same difficulty goes to Possibility 3. Root Ones Ones (+10) Tens Hundreds Root (tier 3 multiplicatives / tier 4 additives with 10s) Ones Ones (+10) Tens Hundreds 0 gloc (use the base roots instead) 0 -oc (use the base roots instead) (add the o at the left) 1 kal gax huu(t) 1 -al -ax 2 mej sup barrel muu(t) 2 -ej -up -arrel 3 gij vers garrel guu(t) 3 -ij -ers -ogarrel 4 ast mult astarrel asuu(t) 4 -(a/e)st -ult -astarrel 5 lun metal lunarrel luu(t) 5 -un -etal -unarrel 6 ferm xenov fermarrel fuu(t) 6 -erm -enov -ermarrel 7 jov hyper jovarrel juu(t) 7 -ov -ohyper -ovarrel 8 sol omniv solarrel suu(t) 8 -ol -omniv -olarrel 9 bet outs betarrel buu(t) 9 -eet / -et -uts -etarrel The limit for Possibility 3 is buubetarreletillion. Possibilities 2 are just milestone proposals, so it is necessary to fill the naming gap up to hutillion. [WIP] (34) truperestillion Possibility 4 is just alternate names for tier 4 “ten” roots, so the 21st Tier-4 -illion with Possibility 4 is idixekalillion. Possibility 5 is just names for 100th and 200th, but it is incomplete. This can be generalized by using the first syllable in Compact Star's hundred roots and removing "r" for additive roots. (826) sobarrermillion Here are the “999” limits for all 5 Possibilities. Underlined means it is changed to match with a respective Possibility. Possibility 1: betanenketillion Possibility 2: bunonuperetillion Possibility 3: buubetarreletillion Possibility 4: bubetixetillion Possibility 5: borbetarreletillion [MADOM POSSIBILITY: 5/27/22] I shout out to DeepLineMedom for making a more reasonable Tier 4 system, with less hyper-cosmological and more Bowersian. The tradition there doesn’t stop abruptly, with an existing language being used like Tier 2 and Tier 3. https://sites.google.com/view/deeplinemadoms-googology/illion-system/first-attempt https://sites.google.com/view/deeplinemadoms-googology/illion-system/remake-2 Note that my new take will now use Madom’s attempts, which I use attempt 1 up to hundreds and remake 2 for hundreds. Also, there are some tweaks to reconstruct my new -illion system, also using Saibian's nomenclature of the form E100{#,#+1,1,2}n. Root Ones Ones (+10) Tens Hundreds 0 gloc (use the base roots instead) 1 kal gax kent 2 mej sup twe deuxent 3 gij vers friel troisent 4 ast mult fioril quaternt 5 lun met finn cinquent 6 ferm xev sexil sixent 7 jov hyp sjour septent 8 sol omni(v) attal huitent 9 bet box neiul neufent Root (multiplied by tier 3) Ones Ones (+10) Hundreds (+tens) 0 -oc 1 -al -ax -ent 2 -ej -up -euxent 3 -ij -ers -oisent 4 -ast -ult -atrent 5 -un -ett -inquent 6 -erm -ev -ixent 7 -ov -ohyp -eptent 8 -ol -omni -uitent 9 -eet -ox -eufent Add “o” to connect with tier 3 multiplicative and tier 4 tens / hundreds root. Add “i” inbetween to construct one-ten combinations. Alternatively, you can also use base-20 with all “one” roots for every hundred. (33) versitweillion is valid, and so gijifrielillion. [CHANGES FROM VERSION 1] Formerly, Tier 3 multiplicatives to -out do not overlay on tier-4 hundreds. It has been changed. Tier 3 Index Name Tier 3 Index Name 1057 outillion 10300 hutillion 21057 dutillion 10600 mutillion 31057 trutillion 10900 gutillion 41057 tutillion 101,200 astutillion 51057 putillion 101,500 lutillion 61057 exutillion 101,800 futillion 71057 zutillion 102,100 sutillion 81057 yutillion 102,400 jutillion 91057 nutillion 102,700 butillion Alternatively, you can keep “o” as it is except hutillion. It’s optional to mind the letters out. The 40th Tier-4 illion was called “aarrillion” before DeepLineMadom made it more contingent. Besides with the connections between units and tens, I split from GeneralStar’s ones. Root Ones Tens 0 (use the base roots instead) 1 -ekal 2 -ej barr- 3 -ij gar[r]- 4 -[e]ast* aarr- 5 -un lunarr- 6 -erm fermar[r]- 7 -ov solarr- 8 -esol jovar[r]- 9 -et betarr- [CHANGES FROM VERSION 2] Anything in Version 2 and prior ones were made from CompactStar’s first version. Root Ones Ones (+10) Tens Hundreds Root (after hundreds) Post-100s Fix 0 gloc (use the base roots instead) 1 tal 1 kal gax hu(t) 2 tej 2 mej sup barr mu(t) 3 tij 3 gij vers gar[r] gu(t) 4 tast 4 ast mult aarr as[t]u(t) 17 typ 5 lun met lunarr lu(t) 18 tomn 6 ferm xev fermar[r] fu(t) 19 tout 7 jov hyp jovar[r] ju(t) 40 tastarr 8 sol omni(v) solarr su(t) 9 bet out betarr bu(t) Root (multiplied by tier 3) Ones Ones (+10) Tens Hundreds 0 -oc (use the base roots instead) (add the o at the left) 1 -al -ax 2 -ej -up -obarr 3 -ij -ers -ogar[r] 4 -ast -ult -oaarr 5 -un -ett -unarr 6 -erm -ev -ermar[r] 7 -ov -ohyp -ovar[r] 8 -ol / -esol -omni -olarr 9 -eet / -et -ut -etarr Use “option 2” if it is followed by tens. Remove (t) if it comes before “tens” / “ones” roots. [r/t] are optional to remove. If there’s a root found in “Post-100 Fix,” replace it if it is followed by a “hundreds” root. [Tier 5] - HOAXORDINARY :[ 10^10^10^10^3000 - 10^10^10^10^10^3000 ]: Tier 5 is the first tier not shown in Bowers’ original list. Let’s keep pushing towards the summit! unvillion = z(1, 5) [1,000th tier-4] I’ll use Sbiis’ system whereas I use “i” connection to add Tier 5 root with Tier 3. (tier 4 roots alone are okay with Fonster’s) Check the original source by Madom. (e.x. unvimeg = 103,000+2nd tier-3) “Thought I’m done with Tier 5 reformations?” Gaps are filled by CompactStar except it’s not doing well. I generalize that “u” is used to add Tier 5 with a Tier 4 root. By word magic, we can remove some letters for shortening reasons. Here’s the multiplicatives. Root Ones (+10) (+tens) Tens 0 glo(c) (unchanged) (unchanged except barr) 1 k ga 2 m su bar(r)[1] 3 g ver gic 4 a mul asc 5 l me luc 6 f xe ferc 7 j hy joc 8 s omn solc 9 b ou bec [1] (r) is required to connect with “+tens” roots. For Tier 3 multiplicatives, add o- inbetween. “Hundred” roots remain unchanged. Anyways, reveal the fifth tier root lookup! Root Ones Ones (+10) Tens Hundreds 0 ek (use “ones” for pre-13) 1 unv ev ek evaf 2 duv twyv lan duvaf 3 et etk etaf 4 cuv cuk cuvaf 5 quiv quik quivaf 6 swiv swik swivaf 7 pryv pryk pyrvaf 8 oiv oik oivaf 9 en enk envaf Tier 4 multiplicatives: [1] For duv, remove “d.” [2] For tier-5 roots that doesn’t start with “u,” add “u” inbetween. (e.x. muekillion = 21030 tier-4 illion = z(21030, 4)) Seeing from the luminosity of magic, the system favors the mistakes. "Knowledge from Sbiis’ hoax prefixes inspires the fix for these roots. From there, the letters were eradicated with some conflict fixes. A magical change, and the roots blew up like an explosion!" /// /// [ NUMBER LIST COMING SOON ] /// /// [Tier 6] - GREECE QUANTITATIVE :[ 10^10^10^10^10^3000 - 10^10^10^10^10^10^3000 ]: CompactStar decided that Tier 6 is based on greek quantitative prefixes. In the greek language: mono / hapax means once, or one times dis means twice, or two times. tris means thrice, or three times. tetrakis means four times. and so on. You can check out the full list at the “numeral prefix” page. This “times” part makes sense by combining tier 5 roots as multipliers to tier 6. For example: “bexatetrakis” means “tetrakis” times “bexa” for me. Because tetrakis is 1012 in tier 5, and bexa is 200, the meaning can be interpreted as 200*1012. Like tier 5, he didn’t make tier 5 roots after the tier 6 milestones. That’s why I will make up additive roots for combining these tiers. Instead of using “e,” I decided to use a greek language of “plus” called “συν.” It’s pronounced as “syn” in English. It will be used for “-is” ending roots only. [subjoin - episynápto] Because there are 4 ending roots for tier 6 names, there will be a simple table below showing the normal/additive roots. Index Normal Additive Recommended 1 [ha]pax pa(ki/ka)- pax/paka 2 -dis d(u/i)sy(n/i/a)- dis/disya 3 -tris trisy(n/i/a)- tris/trisya 4+ -kis (ki)sy(n/i/a)- kis/sya [1] [ha] should be removed when it precedes with Tier 5. [2] You can choose any option in respective () roots. Here’s a table lookup for tier-6 -illions, because it is not difficult to fill up the roots beyond 20. Index Base Ones Ones (+10) Tens Hundreds 1 pax/paka hena hendeca hecatonta 2 dis/disya dya dodeca icos(a/i) diacos(a/i) 3 tris/trisya trya decatria triaconta triacos(a/i) 4 tetra decatetra tessaraconta tetracos(a/i) 5 penta decapenta penteconta pentacos(a/i) 6 hexa decahexa hexeconta hexacos(a/i) 7 hepta decahepta hebdomeconta heptacos(a/i) 8 octa decaocta ogdoeconta octacos(a/i) 9 enna decaennea eneneconta enneacos(a/i) 10 deca For naming tier 6: If x < 4, then only use the base names. Use the second option if tier 5 is before it. Do not use the tens column, if x mod 100 is less than 20. Instead, use the second “ones” column for roots between 10th and 20th. If there is (a/i) option, then use a if there isn’t a tier 6 root before you. Elsewise, use i. Finally, add -akis at the end. Use -asya instead if tier 5 is before it. That’s it! Now, here is an overview table for tier-5 -illions beyond hapaxillion. Number Tier 5 Index Tier 6 Index Name Alt. H5#1,000 1,000 1 hapaxillion H4#(103,000+1) 1,000+ 1+ hapaxe-hepillion H4#(2103,000) 1,000 meje-hapaxillion H5#1,001 1,001 hapaka-hepillion hapa’hepillion H5#1,002 1,002 hapaka-ottillion hapa’ottillion H5#1,010 1,010 hapaka-geopillion H5#1,100 1,100 hapaka-alphillion H5#2,000 2,000 1* ottapaxillion otig’paxillion H5#2,001 2,001 otig’paka-hepillion ottapaka-hepillion otig’pa’hepillion H5#3,000 3,000 neatapaxillion neg’paxillion H5#10,000 10,000 geopapaxillion geg’paxillion H5#11,000 11,000 gea’hepapaxillion geg’heg’paxillion geg’hepapaxillion H5#11,001 11,001 gea’heg’paka-hepillion H5#20,000 20,000 amosapaxillion amg’paxillion H5#100,000 100,000 alphapaxillion alg’paxillion H5#1,000,000 1,000,000 2 disillion H5#1,000,001 1,000,001 2+ disya-hepillion dusyn’hepillion H5#2,000,000 2,000,000 2* ottadisillion otig’disillion H5#2,000,001 2,000,001 otig’disya-hepillion H5#10,000,000 10,000,000 geopadisillion H5#100,000,000 100,000,000 alphadisillion H5#1,000,000,000 1,000,000,000 3 trisillion H5#1,000,000,001 1,000,000,001 3+ trisyi-hepillion trisyn’hepillion H5#1,000,000,000,000 1,000,000,000,000 4 tetrakisillion H5#1,000,000,000,001 1,000,000,000,001 4+ tetrasyi-hepillion tetrakisyn’hepillion Tier 6 Index Name H6#5 pentakisillion H6#6 hexakisillion H6#7 heptakisillion H6#8 octakisillion H6#9 ennakisillion H6#10 decakisillion H6#11 hendecakisillion H6#12 dodecakisillion H6#13 decatriakisillion H6#14 decatetrakisillion H6#15 decapentakisillion H6#20 icosakisillion H6#21 icosihenakisillion H6#22 icosidyakisillion H6#23 icositryakisillion H6#24 icositetrakisillion H6#25 icosipentakisillion H6#30 triacontakisillion H6#31 triacontahenakisillion H6#40 tesseracontakisillion H6#41 tesseracontahenakisillion H6#50 pentecontakisillion H6#60 hexecontakisillion H6#70 hebdomecontakisillion H6#80 ogdoecontakisillion H6#90 enenecontakisillion H6#100 hectatonakisillion H6#101 hectatonahenakisillion H6#110 hectatonadecakisillion H6#111 hectatonahendecakisillion H6#120 hectatonaicosakisillion H6#200 diacosakisillion H6#201 diacosihenakisillion H6#300 triacosakisillion H6#400 tetracosakisillion H6#500 pentacosakisillion H6#600 hexacosakisillion H6#700 heptacosakisillion H6#800 octacosakisillion H6#900 ennacosakisillion H6#990 ennacosienenecontakisillion H6#999 ennacosienenecontaennakisillion [OLD TIER 5 MULTIPLICATIVES] At this point, precedent roots are replaced, if directly followed by tier 6 then tier 5. Index Ones Tens Hundreds 1 heg’ geg’ alg’ 2 otig’ amg’ beg’ 3 neg’ hag’ gag’ 4 deg’ kyg’ deg’ 5 ung’ pig’ theg’ 6 eng’ sag’ iog’ 7 fig’ preg’ kag’ 8 syg’ nig’ lag’ 9 brog’ zog’ sig’ (example: otig’paka-ottillion = 2,002nd Tier 5) (example: sigma’nig’disya-hapaxillion = 980,001,000th Tier 5) We already entered the Tetrational class, surpassing Class 6! It would be the best stopping point for this document. The main part is over. Feel free to go somewhere, or click the links below for curiosity. Naming 10^^n numbers Extended Standard Sheet Overview DeepLineMadom’s Alternate MrRedShark77’s Continuation Aarex’s Googology “Onto the journey of endless stairs of googology.“ “Thy Eternal, Endless of Zillions.“ [[ SPECTRAL REALM ]] Welcome to the realm of tetrational! Exponents now feel close between 2 floors, thus aren't significant. We still feel like simpler formulas need special treatments from illions, hence low Tier 1. [Tier 7] - HEPTACHROMATIC :[ 10^10^10^10^10^10^3000 - 10^10^10^10^10^10^3,000,000 ]: [CHECK OUT NEW ILLIONS RIGHT] Redillion = 1,000-th tier 6 -illion. (That’s 1,000^^7*1,000!) (not by me, but it is found somewhere around 2020 - 2021) But I am not done yet. I need to combine it with tier 6 multipliers / additives to finish this. Tier 5 additives: Add “is” on the right; the option is the “sy” part! Tier 6 additives: Change “red” with “resi” or “redona,” and replace the base roots below. 1: enkapax 2: dyakis 3: tryakis Tier 6 multiplicatives: Just replace the final tier 6 roots with specific roots below: Index Ones Tens Hundreds 1 heke decise hecatise 2 dyke icoise diacise 3 tryke triacone triacise 4 tetrke tessaracone tetracise 5 penke pentecone pentacise 6 hexke hexecone hexacise 7 eptike hebdomecone heptacise 8 oceke ogdocone octacise 9 eneke enenecone enneacise No, there aren’t base roots for tier 6 multiplicatives. Tier 6 Index Name Tier 6 Index Name 1,000 redillion 2,000 dyke-redillion 1,000+ redis-hapaxillion 2,000+ dyke-redis-hapaxillion 1,000* ottaredillion 2,001 dyke-resi-ekapaxillion 1,001 resi-ekapaxillion 2,010 dyke-resi-decakisillion 1,001+ resi-ekapa-hepillion 3,000 tryke-redillion 1,002 resi-dyakisillion 4,000 tetrke-redillion 1,002+ resi-dyasyn-hepillion 5,000 penke-redillion 1,003 resi-tryakisillion 6,000 hexke-redillion 1,004 resi-tetrakisillion 7,000 epitke-redillion 1,005 resi-pentakisillion 8,000 oceke-redillion 1,010 resi-decakisillion 9,000 eneke-redillion 1,011 resi-hendecakisillion 10,000 decise-redillion 1,012 resi-dodecakisillion 11,000 decaheke-redillion 1,013 resi-decatriakisillion 12,000 decadyke-redillion 1,020 resi-icosakisillion 20,000 icoise-redillion 1,021 resi-icosahenakisillion 21,000 icosaheke-redillion 1,030 resi-triacontakisillion 30,000 triacone-redillion 1,100 resi-hectatonakisillion 31,000 triacontaheke-redillion Tier 6 Index Name 40,000 tesseracone-redillion 50,000 pentecone-redillion 60,000 hexecone-redillion 70,000 hebdomecone-redillion 80,000 ogdocone-redillion 90,000 enenecone-redillion 100,000 hectise-redillion 101,000 hectatonaheke-redillion 110,000 hectatonadecise-redillion 200,000 diacise-redillion 201,000 diacosiheke-redillion 300,000 triacise-redillion 400,000 tetracise-redillion 500,000 pentacise-redillion 600,000 hexacise-redillion 700,000 heptacise-redillion 800,000 octacise-redillion 900,000 ennacise-redillion 990,000 ennacosienenecone-redillion 999,000 ennacosienenecontaeneke-redillion [Tier 8] - … Due to the randomness, Cloudy176's faketest won't be included in Tiers 5+. [The Finale] - INFINITE HORIZON The plan for this chapter is to make an arbitrary customizing scheme with adjustable tiers. To do this, we need a new format covering 3 attributes, and 1 additional attribute: Tier. “Tier” means it is a layer from the power tower of 1,000s. We name Tiers as greek numeral prefixes but ending with e instead of a, except that Tier 7 is re, not hepte. Now, we are settling off from the superexponential range! Coming onto the normal attributes, we need a custom one. We observe “d” after “re,” so it is assigned as 1 in a tier coefficient. Thus, octedillion is 1st tier-8 illion, around H8#1. For 2nd tier-7 -illion, I would call this redualillion, making “dual” as coefficient 2 -affix. It feels like a mutant off from the start. Then, we can use the Greek number prefixes with additional “-as” ending. This completes the naming gap between spectral tiers. (ennal = 9x, pentaicosal = 25x) Here are tier 7 milestones. Tier 7 Index Name H1#7 redillion H2#7 redualillion H3#7 retrialillion H4#7 retetralillion H5#7 repentalillion H6#7 rehexalillion H7#7 reheptalillion H8#7 reoctalillion H9#7 re-ennealillion H10#7 redecalillion H11#7 rehendecalillion H12#7 redodecalillion H13#7 retridecalillion H14#7 retetradecalillion H15#7 repentadecalillion H16#7 rehexadecalillion H17#7 reheptadecalillion H18#7 reoctadecalillion H19#7 reennadecalillion H20#7 reicosalillion H21#7 rehenicosalillion H30#7 retriacontalillion H40#7 retetracontalillion H50#7 repentacontalillion H60#7 rehexacontalillion H70#7 reheptacontalillion H80#7 reoctacontalillion H90#7 re-enneacontalillion H100#7 rehectalillion For Tier 6 additives: we have “resi,” so how about we do the same? Just replace “al” with “asi” instead! Tier 6 Index Name H1,001#6 resi-enkapaxillion H1,000,001#6 reduesi-enkapaxillion H1,000,000,001#6 retriesi-enkapaxillion And now, the moment of “infinitum.” For tier-x multiplicatives, remove “l” from “al.” ([tier prefix][quantitative value x]+”a” = *x to Tier-[tier] value) For tier-x additives, replace “al” with “e” + [tier prefix] (“e”+[tier prefix][quantitative value x] = +x to Tier-[tier] value) If there’s “ed” instead of “al,” use “ae” + [tier prefix] If we add a root as a tier-n additive, but not at tier-n originally, add “e” + [tier-n prefix]. Phew, I’m now done climbing the infinite ladder! Value Name H1#7 redillion H103,000+1#5 redis-hepillion H103,000+1,000#5 redis-hapaxillion H1,001#6 resi-enkapaxillion H2,000#6 dyke-redillion H10,000#6 decise-redillion H100,000#6 hectise-redillion H2#7 redualillion H1,000,001#6 reduasi-enkapaxillion H10,000,000#6 decise-redualillion H3#7 retrialillion H10#7 redecalillion H100#7 rehectalillion H1#8 octedillion H103,000+1#6 octesi-enkapaxillion H103,000+1,000#6 octesi-redillion H1,001#7 octae-redillion H2,000#7 redua-octedillion H10,000#7 redeca-octedillion H100,000#7 rehecta-octedillion H2#8 octedualillion H1,000,001#7 octedue-redillion H10,000,000#7 redeca-octedualillion H3#8 octetrialillion H10#8 octedecalillion H100#8 octehectalillion H1#9 ennedillion H103,000+1#7 ennae-redillion H103,000+1,000#7 ennae-re-octedillion H1,001#8 ennae-octedillion H2,000#8 octedua-ennedillion H10,000#8 octedeca-ennedillion H100,000#8 octehecta-ennedillion H2#9 ennedualillion H1,000,001#8 ennedue-octedillion H10,000,000#8 octedeca-ennedualillion H3#9 ennetrialillion H10#9 ennedecalillion H100#9 ennehectalillion H1#10 decedillion H103,000+1#8 decae-octedillion H103,000+1,000#8 decae-octe-ennedillion H1,001#9 decae-ennedillion H2,000#9 ennedua-decedillion H10,000#9 ennedeca-decedillion H100,000#9 ennehecta-decedillion H2#10 decedualillion H1,000,001#9 decedue-ennedillion H10,000,000#9 ennedeca-decedualillion H3#10 decetrialillion H10#10 decedecalillion H100#10 decehectalillion And so on… Imagine the infinite tiers of illions! Now, the imagination came true. Mind-blowing! Lightning’s out! You can check out my other naming numbers: 10^^x numbers Hyper-E Numbers Extended Standard Sheet Overview DeepLineMadom’s Alternate MrRedShark77’s Continuation Trialogue = 10^10^10 Tetralogue = 10^10^10^10 Pentalogue = 10^10^10^10^10 .... Dekalogue = 10^^10 Hectalogue / Giggol = 10^^100 Chilialogue / Giggolchime = 10^^1000 Tier 7: Redillion Bitterswillion Vermillion Orangillion Goldillion Yellillion Limillion YGillion Greenillion Aquillion Cyanillion Tealillion Blueillion Indigillion (Maoillion) Purpillion Violillion Tramaoillion Tetramaoillion Pentamaoillion Hexamaoillion Heptmaoillion Octamaoillion Ennamaoillion Decamaoillion Magentillion Pinkillion Blackillion Whitillion Greyillion Silvillion Skinillion Brownillion Handillion Wigillion Switillion Orangitillion Tier 8: Erkillion Alejandrillion Bazillion Bethillion Bagillion Bridgillion Codillion Courtillion Devillion Duncillion Evillion Ezekillion Geofillion Gwenillion Harolillion Heathillion Jojolion Tier 9: Izzillion Justillion Katillion Leshawnillion Lindsillion Noahillion Owenillion Sadillion Sierrillion Trentillion Tylillion Derfillion Brittillion Vocillion Proudillion Trainillion Painillion Babillion Scillion Troptillion Bitillion Mitillion Koptillion Hiptillion Dayillion Ontillion Chirpillion Netillion Tier 10: Meakoowillion Wookipillion Coopillion Morefillion Royardillion Troyardillion Tetroyardillion Pentoyardillion Hexoyardillion Heptoyardillion Octoyardillion Ennoyardillion Hogillion Icosoyardillion Pentacontoyardillion Tier 11: Mercurillion Venusillion Earthillion Marsillion Jupitillion Saturnillion Uranillion Neptunillion Plutillion Sunillion Erisillion Dysnomillion Makemakillion Haumillion Gijebrillion Asebrillion Lunebrillion Fermebrillion Jovebrillion Solebrillion Betebrillion Glocebrillion Gaxebrillion Supebrillion Versebrillion Multebrillion Pyrebrillion Guntebrillion Kentrebrillion Onlebrillion Paptrebrillion Housebrillion Trongebrillion Batebrillion Handrebrillion Mastebrillion Gotebrillion Kortebrillion Nenkebrillion Janebrillion Unillion Deuxillion Troisillion Quatrillion Cinqillion Sixillion Sepillion Huitillion Neufillion Dixillion Onzillion Douzillion Treizillion Quatorzillion Quinzillion Seizillion Dixsepillion Dixhuitillion Dixneufillion Vingtillion Trenillion Quarantillion Cinquantillion Soixantillion Soixantedixillion illion Quatrevingtedixillion Cenillion Deuxecenillion Troisecenillion Quatrecenillion Cinqecenillion Sixecenillion Sepecenillion Huitecenillion Neufecenillion Milleillion Hugillion Bigillion Largillion Enormillion Superiorillion Nuclearillion Destructillion Tier 12: Doxillion Hevillillion Traminfillion Goillion Formalillionantillion Jovillillion Inphixillion Cillillion Fagillion Begillion Mineillion Robloxillion Gabillion Jillillion Hexillion Hextillion Seemillion Vagillion Vokillion Traktillion Satrillion Trafllion Centrillillion Rekillillillion Keitillion Formillion Hapillion Frannillion Moxantrajantramaoillion Krisillion MrTopFillion Dolphillion Thomillion Kaffwillion Ion Tier 13: Transillion Fakeillion Triktillion Amsterdillion Minionillion Triacosillion Tetracosillion Pentacosillion Hexacosillion Heptacosillion Octacosillion Ennacosillion Dumillion Gastroadillion Tromillion Omegillion Megamillion Hexttillion Xetillion Vockillion Yoctrikillion Multrillion Elongillion Houzillion Howillion Skyscraperillion Americillion Citillion Carillion Busillion Truckillion Planillion Helicoptrillion Tsarillion Slipillion Tramillion Franillion Jogillion Mocillion Moxillion Moxhamillion Hammerillion Malillion Contillion Dontillion Gontillion Fontillion Hontillion Kontillion Korillion Zerillion Killinillion Hogtillion Seperillion Gamillion Hadramillion Xandrillion Dentillion Bentillion Gazillion Squillion Bajillion Bazillion Jillion Zillion Haxillion Dakkillion Ikkillion Trakkillion Googolillion Oldillion Sexamaoianamdanabsolutedestructillion Hextramillion Heptillion Hopillion Daycarillion Schoolillion Collegillion Ayillion Summerillion Autumnillion Winterillion Springillion Hydroxillion Hotyillion Meccillion Xerillion Autillion Mansionillion Richillion Chinaillion Amazillion Farmillion Trulillion Bulkillion Containerillion Coachillion Idillion Fragmentillion Denptrillion Dentistillion Docterillion Torillion Tarillion Jerillion Detectillion Srirachillion Hamsterillion Tierillion Zackillion Kyrieillion Kattillion Royillion Jansenillion Jerlillion Lorelynillion Maggillion Janeillion Einstillion Kirbillion Eithillion Yocillion Yogillion Turillion Turbillion Oillillion Housalinatadenctahatramaoillion Cillion Honterillion Tier 14: Swimillion Censorillion Villillion Kaltillion Alfillion Rebelaxillion Kathillon Hillillion Fillion Sillion Jillion Zillion Tragicillion Allergicillion Glosillion Gillillion Gradillion Fertillion Daykillion Cillillion Meggillion Giggillion Killillillion Millillillion Gentillion Gintillion Frecklilllion Tractorillion Tortillion Tartillion Nukillion Tortillillion Pieillion Fryillion Xenoxenillion Quintrillion Hepttillion Sotrandillion Fugillion Fungusillion Boxillion Tragillion Alpillion Alkalillion Violinillion Guitarillion Pianillion Organillion Gunillion Paperillion Darkillion Papillion Hurtillion Jakillion Hexamaisadennisianadentamaoillion Exersisillion Trackillion Ickillion Dackillion Micillion Mousillion Dogillion Ramaxillion Ramaxonillion Grendillion Tiehillion Trillillion Pillillion Pillowillion Ice-Creamillion Dogfianillion Treeillion Woodillion Factrillion Sembillion Tamborillion Rimcktillion Aujillion Fijillion Vectrillion Dectillion Fyamillion Hexttillion Textillion Texillion Mexillion Mexicillion Frecklillion Jinillion Missillion Masillion Idrillion Freckillion Astrillion Jovendiramaoillion Gehillion Bukillion Ukrillion Xunillion Verillion Merrillion Aardillion Bravillion Charlillion Delphillion Primillion Obvillion Erkamillion Erillion Micramaoillion Heptanahadrillion Homillion Gonillion Heroillion Policillion Catillion Farmerillion Buildingillion Ckamnillion Damillion Novillion Novamosillion Bosnillion Hermillion Centrifugillion Dransillion Frecklillion Tria-gentillion Tetra-gentillion Penta-gentillion Hexa-gentillion Hepta-gentillion Octa-gentillion Enna-gentillion Triaillion Tetrillion Pentillion Hexxillion Hepptillion Occtillion Ennillion Deccillion Fermmillion Xendakultamultiamaoillion Sevillion Absillion Charlesillion Sectionillion Ikurtansadaurtanikillafreckillion Trakentramaillion Maillillion Jectillion Mectillion Fermentillion AumSumillion MrBeastillion Indianillion Lungillion Liverillion Heartillion Humanbodyillion Gorillaillion Elephantillion Rhinocerosillion Zebraillion Hamsterillion Caribouillion Dogillion Catillion TRexillion Triceratopsillion Crocodileillion Deinosuchusillion Flowerillion Treeillion Notoillion Ncvrillion Humnaillion Ultrakillion Tier 15: Frecklillion Jectillion Yocktillion Xonnillion Zeptrillion Heptrillion Hextrillion Sotillion Rimtillion Troctillion Hentillion Frankillion Itillion Mitchelillion Gogillion Gidgillion Chloeillion Adamillion Jodieillion Heptahapaxenultaquinquagintillion Xannillion Xendillion Globillion Firmillion Fumillion Ferillion Zerillion Millinillillion Tragillion Alfamartillion Malillion Gradillion Grahillion Bukuwillion Scrillion Quinquagintramillion Cendillion Kentillion Ventillion Violatillion Redsillion Legillion Atillion Roysillion Pebblillion Hectyillion Tirillion Ramillion Vigillion Classillion Astrillion Trudillion Harmonillion Octamafrandatechnoiadtyantylillion Villillion Classicalillion Buxillion Bulillion Penillion Pennyillion Denchardillion Benchardillion Yocillion Mantillion Barbillion Quecktosamasteanasotredagentillion Nonagillion Gillillion Fillinillion Killinillion Irillion Ibillion Sodrillion Sexeragentrantagstarastarharlianuatrucentanamillinillion Xenultillion Raxillion Ranaxillion Ranaxonillion Tetrisillion GTAViceCityillion Jocillion Vacillion Pentachillion Yectillion Yulillion Moldovillion Roanillion Transantanillion Xenillion Syrupillion Techillion Lenillion Mohillion Ukillion Proudillion Hondillion Centauillion Xenillion Towerillion Tiredillion Sakllion Togillion Trailillion Fuyillion Huyillion Dectillion Charlieillion Delphinillion Quarantillion Bukuwahillion Cosmeticillion Youtubillion Bankruptillion Violinillion Vigillion Trigillion Quadragillion Quinquagillion Sexagillion Septuagillion Octogillion Nonagillion Centrillion Millunadecamillion Grangillion Millibadratrahadabukuwahadatreeantarillion Absillion Agonillion Panickillion Heppillion Boogillion Boovillion Breachillion Hapexillion Yetillion Motherillion Gayillion Millillillillillillillion Barkillion Logillion Ultrillion Volcillion Infillion Gillillion Kansasillion Susillion Anusillion Thetapillion Novagillion Vagintillion Housajillion Gayajdahamokilacuillion Longdogonillion Ashortleashillion Tier 16: Javillion Freyillion Ukrainilion Hetillion Killinillinillion Maxillion Thowillion Millionillion Ownillion Kerillion Otterillion Macillion Mcdonaldsillion Vaginillion Hempillion Hectarillion Giggillion Treehillion Rillillion Zerkillion Mothillion Vajillion Lahillion Samillion Mallgillion Houseillion Gringaintanyatagentillion Gringillion Oktillion Otillion Playillion Daycillion Grassillion Assillion Xillion Grasshopperillion Hextamillion Hextamillion Sosillion Seoulillion NYCillion LAillion Knowillion Vectillion Cosillion Kisillion Trivillion Gantillion Gangstrillion Gangsterillion Gagillion Missolillion Mossolplexillion Jakeillion Atticillion Lightillion Aarhillion Meakillion Arbillion Icillion Polillion Vifillion Dectillion Gecillion Centyillion Dubillion Vecxillion Becxillion Mecxillion Hecxillion Gentrillion Generillion Vocalillion Obillion Violillillion Nomillion Oilillion Dachillion Xenoxenoxillion Cirillion Cerillion Millingillion Vogillion Trammyillion Mectrillion Ducillion Heroinillion Dakanadranameganamegillion Giftillion Logueillion Trigillion Tetragonillion Pentagonillion Hexagonillion Heptagonillion Octagonillion Ennagonillion Hecillion Kintrillion Mecgillion Otherillion Maxtillion Deckillion Treenillion Quinvigeranillion Giggolillion Herbavrollion Tienillion Hextamillion Pentirillion Pyramidillion Deflillion Throwillion Berillion Tardillion Erillion Pepillion, Peptrillion Oxillion Oxfordillion Kermillion Billimafranaxeantarahapaxennillion Xenoxenoxenillion Denchillion Denctillion Dragonillion Limixillion Radillion Morbillion Megamindezearazeusillion Invillion Januarillion Trigrilantrillion Hillillillion Limousinillion Vagin-Verillion Satillion Sotillion Tier 17: Pentillion Pent-Million Pent-Billion Pent-Trillion Pent-Decillion Pent-Vigintillion Pent-Googol Pent-Quadragintillion Pent-Centillion Pent-Millillion Pent-Myrillion Pent-Micrillion Pent-Googolplex Pent-Hectillion Pent-Killillion Pent-Ikillion Pent-Trakillion Pent-Multillion Pent-Hepillion Pent-Augillion Pent-Hapaxillion (also known as Italillion) Pent-Disillion Pent-Trisillion Pent-Tramaoillion Pent-Decamaoillion Pent-Sotillion Pent-Vagin-Verillion Pent-Charlillion Pent-Tramillion Limitillion Fujillion Hujillion Gujillion Bujillion Dujillion Mujillion Nujillion Limmillion Lyrillion Myrillionmyrillion Moonillion Tiredillion Herillion Erckillion Megacillion Googolplexillion Citillion Teslillon Unteslillion Duoteslillion Triteslillion Quattuorteslillion Quinteslillion Sexteslillion Septemteslillion Octiteslillion Novemteslillion Miragillion Rayillion Jocillion Idyeavocalillion Lillion Trillillillillion Thetapaxillion Xeroxillion Sexillion Vexillion Ravillion Villagerillion Chickillion Pigillion Technoblillion Elillion Vagin-Tramaoillion Verillion Aaronillion Arghillion Brillion Vesillion Gevillion Pingillion Ngerillion Nigillion Molillion Tranquillion Pentyeatrartaratreagentatrigentillion Chavillion Oblivillion Infinitillion Finchillion Triacillion Tetracillion Pentacillion Hexacillion Heptacillion Ocktacillion Ennacillion Dekillion Ickosillion Triacontacillion Tetracontacillion Pentacontacillion Hexacontacillion Heptacontacillion Octacontacillion Ennacontacillion Hectacillion Chillacillion Myriacillion Taxillion Trucillion Trusillion Rustillion Mistillion Kallillion Harlillion Funkillion Dojillion Platrillion Illionillion Ilillion Lovillion Garrayantaragayillion Afragarearafillion Absolutillion Sextratailllion Grisillion Strillion Ikyokillillion Nonecxaxenultillion Tiramisillion Tridekillion Trafillion Ricillion Tier 18: Freakillion Outsidillion Orckillion Googillion Gagillion Gobillion Oblivillillion Tracyillion Freudillion Eudillion Udillion Nobillion Conanillion Doraemillion Novillillion Hexahedrillion Novemdreagarearraradrattayectagintillion Cintillion Ducintillion Trucintillion Quadringintillion Quingintillion Sescintillion Septingintillion Octingintillion Nongintillion Tranquintanovemgaentaedaorantacxentutrambutragayillion Ficnillion Demillion Mellillion Plantillion Animalillion Bacterillion Durillion Danchillion Subillion Ganchillion Ranchillion Rantillion Banchillion Fanchillion Kanchillion Knowillion Swillion Demillion Octataratreagetratratrertrakantratramaintagentillion Delphinillion Walillion Owillion Daneillion Lillintrantamillion Bentlillion Grahillion Bukuwahantramillion Piccillion Prillion Drantillion Kyrillion Kyr-Million Kyr-Billion Kyr-Decillion Kyr-Vigintillion Kyr-Centillion Kyr-Millinillion (Millillion) Kyr-Myrillion Kyr-Micrillion Kyr-Triacontillion Kyr-Hectillion Kyr-Megillion Kyr-Trakillion Kyr-Disillion Kyr-Tramaoillion Kyr-Decamaoillion Kyr-Erkillion Kyr-Kyrillion Dukyrillion Dukuwillion Astlillion Dohectakisillion Chillakisillion Myriakisillion Audillion Vunduktillion Untillion Bentrizillion Troktillion Sotrillion Rimktillion Quektillion Ocktillion Googolnoninplexithrongillion Tollillion Grillion Ennayvunduktsarbombrabtratrakengentillion Enviousillion Encikillion Illishoikillion Mercktaretrathrongethreatreantagayolintramillion Nonexillillion Goxillion Cosillion Garrel-million-trucentillion Gecrtrillion Aardillion Myriamaoillion Chinillion Premiumillion Satillion Nonecxentramaoillion Centytramillion Tier 19: Charlillilllillillion Orcantardillion Foxxillion Millinamaoilion Airlarillion Megamaoillion Sextraudwanuratyliatramaoillion Nancillion Unitedillion Unitillion Freshillion Phillion Goxxillion Ovillion Nonexenultraeuntraillion Conanillion Nobellillion Safeillion Giornoillion Pentaconacontillion Hectamatrillion Cevillion Altaccfillion Jinglillion Grillillion Parillion Mertillion Boovillion Boogollillion Dakotillion Ictreayttartruetranatwakillion Shapillion Hooverrillion Daarrillion Dhaarmannillion Dhenillion Whillion Huntrillion Hutaftraillion Atmillion Gijamsntatillion Darrellillion Barrellillillion Foxintarillion Torendillion Metallillion Omnivorousillion Sandpillion Maccillion Farrillion Harxillion Sortrillion Gogillion Giigillion Terramillion Tesserillion Ggtatartueillion Guweuiequiqyillion Gruelillion Villion Deckillion Unveilillion Gogologueillion Plexillion Descaonudustuntratambillion Plexidingillion Chimillion Berjillion Ferjillion Terjillion Gerjillion Derjillion Nerjillion Firjillion Gingivitillion Ingillion Oncillion Gaxenillion Xeroxeroxeroxoillion Megrillion Gigrillion Terrillion Pettillion Exetillion Zetillion Yotillion Xen-illion Drakillion Irakillion Trokillion Ponininijillion Mijillion Ferillion Asillion Contamaoillion Pentacontrilatrareqtreatrwacxentaramaatranatatracontillion Contyillion Hassolugulillion Sextramajuewyillion Cxillion Environmenillion Housallillion Ratrillion Sutillion Sugillion Sugarrillion Farmillion Hextrantacontillion Coneillion Cylindrillion HexagonalpriGogsmillion Cubillion Spherillion Supercalifragilsticexpialidociousillion Teachillion Cityillion Hapaxillillion Disillillion (Isillion) Jurillon Ovillion Hoxillion Hoverrillion Narrillion Hapacksillion Itallillion Australillion Canadillion Moldillillion Turkmillion Hillion Bhutillion Wuntillion Wanillion Sagannillion Annuntiraillion Naenillion Tier 20: Trektillion Untrektillion Duotrektillion Tretrektillion Quattuortrektillion Quintrektillion Sextrektillion Septingtrektillion Octotrektillion Novemtrektillion Decitrektillion Centitrektillion Ducentitrektillion Trucentitrektillion Quadringentitrektillion Quinquagentitrektillion Sexentitrektillion Septingentitrektillion Octingentitrektillion Nongentitrektillion Millitrektillion Decimillitrektillion Centimillitrektillion Micritrektillion Nanitrektillion Picitrektillion Femtitrektillion Attetrektillion Zeptitrektillion Yoctitrektillion Xonitrektillion Vecitrektillion Mecitrektillio{d}xIcositrektillion Triacontitrektillion Tetricontitrektillion Hectitrektillion Killitrektillion Megtrektillion Dutrektillion Trutrektillion Tetretrektillion Pentetrektillion Hexetrektillion Heptetrektillion Octetrektillion Ennetrektillion Dekatrektillion Gorillion Horserillion Giggillion Jamesillion Sotredraaoillion Doedillion Trakkanayillion Yillion Holillion Bolillion Galillion Myriakisillion Hextramolillion Mejantillion Aquaillion Aidillion Trentonillion Tentillion Gaytillion Hexttextillion Mextillion Pentaamaysyeayeweuwantaranaatcontillion Dark-Trillion Ogillion Verserillion Demillion Gexillion Vonillion Burgerillion Hamburgerillion Unvillion Gloxillion Gaexillion Absillion Transeranctahectrakiasexilion Pentacthulillion Goghoustillion Clacillion Astronomillion Goxygennillion Elonmuskillion Blueyillion Tellillion Goxantranaterillion Lineatrillion Empillion Gigillillillion Thantillion Housamaillion Xillinatmillion Atmosphillion Xevillion Inillion Aumstillion Grocillion Goxianillion Hossillion Mansillion Transfillion Roptillion Tropicillion Burundillion Lesothillion Malawillion Spainillion Americaillion Unitedkindomillion Francillion Québecillion Ontarillion Albertillion Manitobillion PEillion Novascotiaillion Newfoundlandillion BCillion Saskatchewanillion Nunavutillion Philippillion Tatillion Moshillion Maggillion Antillion Robiillion Dionillion Kurtillion Fumillion Xillion Unveilillion Gozzillion Goexillion Meakillion Wookillion Copillion Royillion Troyillion Tetroyillion Pentoyillion Hexoyillion Heptoyillion Octoyillion Ennoyillion Dekoyillion Icosoyillion Pentacontoyillion Hectoyillion Tramaoillillillion Pentamiaduwydwenxyiryiwillion Xyillion Shortdogonillion Alongleashillion Tier 21: Voxillion Duhillion Uhandyillion Atillion Miccillion Nannillion Piccillion Femmillion Atttillion Zepillion Yotillion Xovillion Vekillion Mekillion Duekillion Icokillion Kusnsillion Kunguklillion Kungillion Augillillion Hexamaoudieudwhisgyxwiygyuxgdiygddywgdxillion Xillaaishaysfyuqfwwfzgargoogolapiwdsumdxiuesmghedillion Hyfqillion Dgillion Etayusillion Drsdtsuwmugnwuxndefgyenxashillion Ysnwmgydguyegudmgeuygfmuryillion Hgyubsantillion Franciajuuezwmymgwuhfdengufnguyndneunxwugdnfeyneetfeygfillion \int \frac{\pi \sin\left(\sqrt{x}\right) \, \mathrm{e}^{\sqrt{x}}}{\sqrt{x}} \, \mathrm{d}x: Bexaillion Trexaillion Tetrexaillion Pentexaillion Hexexaillion Heptexaillion Octexaillion Ennexaillion Familyillion Babyillion Boyillion Girlillion Manillion Womanillion Dohillion Ohamillion Japanillion Freewillion Teptillion Violillillion Fillion Erkamillion Bitterillion Swillion Orillion Lillion Aqillion Blillion Indgillion Pupillion Villion Tamillion Rillion Dexiajyioillion Oioioillion Hapakillion Hartillion Meanillion Woopillion Cpillion Moreillion Ictillion Noctosillion Nonosillion Golvillion Vaxillion Hexialillion Rkillion Killillillion Vecekillillillion Hectekillillillion Megamegillion Micryamoillion Partillion Blueintaraiuquaajnaahsuauasganmanuaardillion Grinchilllion Santillion Holillion Buctillion Ogdoeillion Loanillion Gugillion Bridillion Lxeoliauaasmgsasgatrigintillion Obviousillion Moserillion Tier 23: Goisillion Gyitillion Itarillion Blilamaoillion Centurionillion Blasphemorgulusillion Treethreeillion Thothillion Tetillion Hesaihsuaillion Royardimieaaiuillion Bluentamillion Gahulaguajiliahuagfillion Biwlwlion Nillion Nonillillillion Millinillinillinillion Kajillion Dajillion Trajillion Tajillion Pajillion Exajillion Zajillion Yajillion Najillion Dakajillion Hotajillion Meghillion Gighillion Astrillion Luntoxamillion Jrettiamillion Gixillion Quinsaayillion Gfillion Hillion Hozoxillion Boxillion Lugillion Missolillion Blasphemillion Centurillion Gongillion Giaoillionajillion Giaillion Aillion Aug-Maxcillion Xcillion Meakoowardillion Wookipardillion Royardyardillion Yardillion Tramillion Xillinillion Millinillillinillinillion Yillillion Fartillion Shinamillion Mantrillion Binocsillion Kittillion Drillion Handriamaquadrillion Goramillion Xianillion Gulillion Goaniamtillion Matillion Mathematicsillion Sciencillion Englilion Filipillion Homeroomillion Apillion Myillion Myopiaillion Phobillion Ganemillion Hogamillion Hovamillion Hoovillion Hoxillion Fbillion Polillion Tolillion Thorillion Supermillion Batmillion Wondillion Willoughbillion Timillion Perfectillion Hoxiuisaosuaahgsdgsyillion Upillion Downillion Loudillion Duhamaillion Ybaillion Libtillion Lubillion Gexillion Goxiasolillion Gasanxillion Crashillion Hyamillion Ammillion Ohajnillion Oniamillion Xillinaillion Hardillion Fortamillion Firmillion Hippillion Vandalisillion Edillion Johnillion Georgillion Geraldillion Cooperillion Haroldillion Derpillion Goalillion Gpillion Pillillillion Ixillion Tramillillion Uabtamillion Ubtillion Ugtillion Uhtillion Untiamillion Ynatillion Dnalillillion Buamayillion Hamillion Hummerillion Toyotillion Decorillion Blueintramaoaillion Baxisoillion Baxillion Oskillion Haosmillion Smaillion Smashillion Hdillion Hdyillion Darrillion Garraillion Absolutelillion Gintamillion Ungentillion Dumnaillion Planyillion Tier 24: Goanaillion Yillion Yamillion Funtamillion Luntamillion Gisillion Glosillion Italyiamillion Italamillion Hoexillion Hamtillion Xillinillion Charillion Delphamillion Intamillion Untillillion Bentrizillillion Troktillillion Sotrillillion Rimktillillion Quektillillion Pektillillion Ocktillillion Vedillion Enosillion Heneosillion Cantaramialalioaannaillion Hexaamillion Gexillion Losillion Greatillion Big-Hossillion Goanimatillion Tramillillillion Tramaoillillillion Dakultarelillion Harellillion Dossillion Moasillion Ossillion Novellillion Decxillion Goxialillion Tranillion Hgaillion Oggillion Mikillion MekillIon Bekillion Trekillion Tetrekillion Pentekillion Hexekillion Heptekillion Octekillion Ennekillion Decekillion Icosekillion Triacontekillion Tetracontekillion Pentacontekillion Hexacontekillion Fintillion Hogamontamantarmackillion Idxillion Joctillion Kosillion Jintramrillion Killillillillillillillion Gargillion Harlillion Govillion Gohillion Buhillion Biotillion Nedillion Gitterswillion Borangillion Holdillion Lellillion Bimillion Leenillion Uquillion Glueillion Vindigillion Burpillion Volillion Yramaoillion Petramaoillion Hentamaoillion Gexamaoillion Jeptmaoillion Poctamaoillion Mennamaoillion Orkillion Blalejandrillion Mizzillion Goahillion Hentamillion Pexillion Gorillion Garillion Harillion Blasphillion Gogoltillion Gexamillion Examaoillion Zettamaoillion Doxamillion Hostamiaauahyaratyfatyafytfauafacillion Fossillion Kostillion Kormillion Kyrieillion Hyrillion Byrillion Fyrillion Gyramillion Augillillillillion Indonesillion Goramillion Dormamillion Xillanillion Mottamillion Kuhillion Kungulillion Gormilllillillillion Axillion Edillion Yoctacontillion Noachillion Champillion Pillinamillion Bintillion Gorikkillion Hortamillion Totiamillion Troktamillion Grandmothillion Grandpillion Pintillintamaoillion Hoanillion Oxamaoillion Yottamaoillion Xennamaoillion Dakamaoillion Ikamaoillion Trakamaoillion Tekamaoillion Pekamaoillion Exakamaoillion Zakamaoillion Yokamaoillion Nekamaoillion Entamillion Glizzyillion Duhamillion Glocillillion Gertamillion Merillion Berillion Trerillion Tetrerillion Penterillion Hexerillion Hepterillion Octerillion Ennerillion Dekerillion Tier 25: Giotillion Animillion Kolillion Dolillion Trolillion Tolillion Polillion Exolillion Zolillion Yolillion Xolilion Groanillion Hortamillion Groacillion Yocxillion Superfragillion Hormillion Rorillion Jocyillion Luckillion Dekaxillion Kaxillion Taxillion Buntamillion Gocillion Hapaxxillion Dissillion Trisquillion Joerillion Joruntillion Toxillion Godillion Cenillion Blasillion Gongulillion Greatillion Gwenamillion Kisillion Hisillion Bisillion Lisillion Tisillion Quillillion Hoeckillion Ckamnamillion Killinamillion Namillion Buamillion Yramillion Petramillion Nentamilion Aexamillion Eptamillion Poctamillion Nennamillion Gillillion Kortanaishuagyscudysifsifgsdysdfgfnuudngdillion Dralillion Kvillion Fvfvfvillion Fvillion Kistmillion Tmanatamaillion Tmaoilnatillion Tillillillillillillillion Concrillion Killinaminataillion Matillion Mortillion Stopillion Goillion Xirtamillion Urtillion Ouchillion Horta-Millinillinillion Yoctamillion Kalamaoillion Kolamillion Bolillion Trolilion Tetrolillion Pentolillion Hexolillion Heptolillion Octolillion Ennolillion Dekolillion Icosolillion Triantolillion Hectolillion Duaardillion Truaardillion Tertruaardillion Pentuaardillion Hexuaardillion Heptuaardillion Octuaardilion Ennuardillion Dekuaardillion Bolillianthillion Mirrillion Raxamillion Noobillon Glockillion Vulillion Tetamillion Tamillion Mejamaoillion Gijamaoillion Astamaoillion Lunamaoillion Fermamaoillion Jovamaoillion Solamaoillion Betamaoillion Glocamaoillion Gaxamaoillion Supermaoillion Versamaoillion Multamaoillion Nonecxenultamaoillion Hoerillion Ertillion Erfillion Wtgamaoillion Tvillion Goaushduqxtillion Gnsydwywgndyillion Yuiwdmdiuillion Jdywjuyillion Ihewygeuyengdefguyeillion Metamaoillion Xevamaoillion Idixamaoillion Hepamaoillion Ottamaoillion Neatamaoillion Goatcillion Troktosauntamillion Stramsotrillion Truktillion Setrillion Romktillion Guektillion Vektillion Pocktillion Ovedillion Novillinamillion Hortillion Dubravillion Trubravillion Tetrubravillion Pentubravillion Hexubravillion Heptubravillion Octubravillion Ennubravillion Dekubravillion Hostamillion Blakillion Cocillion Vettamillion Burjillion Pyramidillion Mountainillion Dahillion Hottillion Deatamaoillion Unatamaoillion Entamaoillion Fitamaoillion Sytamaoillion Brontamaoillion Geopamaoillion Alphamaoillion Hapaxamaoillion Decakisamaoillion Redamaoillion Tramaomaoillion Tetramaomaoillion Pentamaomaoillion Hexamaomaoillion Heptmaomaoillion Octamaomaoillion Ennamaomaoillion Decamaomaoillion Ingamillillion Erkamaoillion Izzamaoillion Meakoowamaoillion Atomamaoillion Humaoillion Staramaoillion Alphamaoillion Duhamaoillion Superfragilicexpialidociousamaoillion Sicillion Tier 26: Houxillion Inphillion Gocillion Hactillion Kallillion Medillion Gishillion Kurillion Crashamaoillion Tramamaoillion Aardamaoillion Exillillion Xintillion Xertamillion Xeenillion Tennillion Tractillion Bockillion Dramillion Astmillion Thantoxillion Ohsuhdgillion Gusauduiwesgudwnillion Sopillion Hepillillion Zetiantillion Tetramamaoillion Xillamaoillion Bravamaoillion Charlamaoillion Delphamaoillion Primamaoillion Obvamaoillion Ducharlillion Trucharlillion Tetrucharlillion Pentucharlillion Hexucharlillion Heptucharlillion Octucharlillion Ennucharlillion Dekucharlillion Icosucharlillion Untamaoaintamillillion Uillion Unsuctillion Duoillion Trillinaillion Dumintillion Sexillion Doxasillion Goxilnyillion Ontamaointillion Maoillinillion Pintillion Tetramaoillillion Hoxamilion Burjillillion Rotillion Tier 27: Icosillinamaoillillillion Illinamillion Namamaillion Royillinatillion Nadfillion Fillion Gortamillion Redsyillion Octacontillinamaoillintamillion Bliuillion Iuiuillion Lucillion Johnillion Jovannillion Gocillion Glaxillion Sthiamillion Bintillion Jinikillion Hoxamaoaianayatamaaoaaiaayajauayakaajayahananantillion Voccillion Tri-billion Hoxrillion Finamillion Finalillion ( Not the Lastillion ) Dakfinalillion Hotfinalillion Uauillion Fvfvillillion Fzgillion Kiltillion Horammillion Grannmamillion Pentamaoillillion Dudelphillion Trudelphillion Tetrudelphillion Pentudelphillion Hexudelphillion Heptudelphillion Octudelphillion Ennudelphillion Dekudelphillion Killinamillion Hamillion Ozzillion Minecrillion Roblillion Hoxsjwhdquxgillion Gdillion Groundillion Osillion Klunatikillion Ayasmarillion Mukbangillion Bridgesillion Tteybokillion Bochillion Bochahendatanushotoridakillion Hoxtillion Aeroplillion Ikultamaakosnxdysanunscillion Tranusxsgxhedcusybdufdsydfsbtssfdaatrainillion Superfragilcillion Gowillillion Klunatikillion Hoxiantillion Duprimillion Truprimillion Tetruprimillion Pentuprimillion Hexuprimillion Heptuprimillion Octuprimillion Ennuprimillion Dekuprimillion Trillinamillion Hoazillion Ghsaxdydgdwuyagusillion Croshillion Croatillion Bosnillion Montenegrillion Kosovillion Albanillion Greecillion Ocdsajovamillion Ogdillion Ooilahillion Tier 28: Hoxddillion Jostamillion Joplillion Roskanillion Jovillinillion Hociclillion Cicillion Rontillion Rostillion Kostillion Morrillion Jokillion Rhinillion Lionillion Tigillion Monkillion Animalillion Jodwillion Hjdixoillion Oixwhufillion Ugsguqwnguwikameakoowillion Kooilion Kookoowillion Meakoowillinillion Hiwikluntillion Destillion Hohohillion Hidsalillion Hoxillanillion Xohillion Xevantadkuntamillion Dkamillion Djamnillion Africillion Jocillion Joswuillion Hortiamillion Kistillion Askillion Hostillillion Planetillillillion Josillion Fosillion Fisyillion Flexillion Fiillion Illion Hoxislilaswuiqsduyillion Killamillion Korillion Jostiamillion Howillion Kostamillion Bismillion Hostmaillion Aillion Airillion Oxillion Ottamaoillillion Extramaoillion Hypermaoillion Hyperalillion Mosilamoillion Oimsillion Nosconillion Duobvillion Truobvillion Tetruobvillion Pentuobvillion Hexuobvillion Heptuobvillion Octuobvillion Ennuobvillion Dekuobvillion Hodamillion Modillion Dkamianmillion Denpentajupentacontoyardillion Aardillillion Bravillillion Charlillillion Delphillillion Primillillion Obvillillion Delphinillion Tarliamillion Liamillion Johamillion Johnnyillion Johnyillion Missolplexillion Admillion Kosiolsmillion Rosillion Kisillion Lillillillion Millillillillillillillion Oblivionillion Utteroblivionillion Infinityillion Omegaillion Epsilonillion Zetaillion Etaillion Phillion Gammaillion Churchillion Mahlillion Weakillion Absolutillion (Absolute Infinityillion) Everyillion Growthillion Endlessillion The-Endillion Voidillion Endillion Stoppingillion Zerillillillion Bondazozillion Tier 29: Gaggolillion Triaillion Kiallmaillion Gigolillion Emperollillion Hoajillion Hozuoillion Moxiamillion Rosmillion Hlozillion Hozizoalaiosjillion Jzozoillion Iaizhziuillion KillinMaoillion Doctillion Millionillion Killillionillion Aardillionillion Joxiamillamnqjsaiqxrhuieanschmittypleasehelpmeillion Jociamtamtrigintramaoillion Bravamaoillion Novillinillion Tier 30: Lelillion Adesresdertillion Dogillion Catillion Horsillion Hospitillion Illionillion Novillion Conditioillion Lillillion Berillion Scgillion Wasillion Ededededededededillion Grangolillion Googologyillion Doogillion Filillion Novillinillinillion Novillinillionillion Sescinyillion Beartillion Erillion Vcillion Trigintillillion Adressillion Eightillion Nineillion Tenthillion Hundredillion *%illion Zooillion Thousandillion Examples using the "H-illion function and -illion name (e.g. 1 quattortrigintillion, 1 micrekilatrecillion) format": H34 = H34#1 = H(1,000^34)#0 = 1,000*1,000^34 = 10^105 = 1 quattortrigintillion H2,013#2 = H(1,000^2,013)#1 = 1,000^(1,000^2,013+1) = 10^(3*10^6,039+3) = 1 micrekillatrecillion Finalillion = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{243}} } } } } } } }+3} Lastillion = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{273}} } } } } } } }+3} Endillion = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{303}} } } } } } } }+3} Limitillion (last of -illions in the list of googologisms in Googology Wiki) = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{3003}} } } } } } } }+3} TREE(3) SSCG(3) Big Boowa = \lbrace 3,3,3 / 2 \rbrace Big Hoss = \{100,100 \underbrace{////\ldots////}_{100} 2\} = \{100,100 (1)/ 2\} Great Big Hoss = \{\text{big hoss},\text{big hoss} \underbrace{////\ldots////}_{\text{big hoss}} 2\} Bukuwaha = \{L,X^X\}_{100,100} BIGG = 200? = 200![[_{<1(200)2>⁅200⁆} 1]] = f_{\psi(\psi_{I_{\omega}}(0))}(200) Goshomity = \{100,100 \underbrace{\backslash\backslash\backslash\backslash\ldots\backslash\backslash\backslash\backslash}_{100} 2\} = \lbrace L2,100\rbrace_{100,100} Meamealokopoowa = \{L100,10\}_{10,10} Meamealokopoowa-Oompa = \{\{L100,10\}_{10,10}\&L,10\}_{10,10} Tritar = Tar(3) Quadritar = Tar(4) Quintitar = Tar(5) ... Dekotar = Tar(10) Hektotar = Tar(100) Kilotar/Chilliotar = Tar(1 000) ... Unintar = Tar(Tar(10)) Bintar = Tar(Tar(Tar(10))) Trintar = Tar^4(10) ... Tarintar = Tar^{Tar}(Tar) Loader's Number = D^5(99) 1919th Busy Beaver = Σ(1919) Fish Number 4 = F_4^{63}(3) 10^6th Xi function number = Ξ(1 000 000) \Sigma_\infty(10^9) Rayo's Number Fish number 7 = F_{7}^{63}(10^{100}) BIG FOOT = \text{FOOT}^{10}(10^{100}) Little Bigeddon Sasquatch Double Sasquatch Large Garden Number = f^{10}(10 \uparrow^{10} 10) Hollom's number = \approx \mathrm{I}_{6.895 \times 10^{104}}(200) Oblivion = K(gongulus) Utter Oblivion = the largest finite number that can be uniquely defined using no more than an oblivion symbols Ultimate Oblivion Hyper Oblivion Sam's Number = \mathrm{Sam}_{\omega_{\varepsilon_{\omega_1^{\mathrm{CK}}\times f_{\psi(K)\times \mathrm{E}100\#100\#100}(\mathrm{Rayo}(10^{100}))}}^{\mathrm{CK}}}(Large Garden Number) Crouton Croutonillion Finity Infinity Aleph Null ω ω+1 ω2 ω^2 ω^ω ε0 ε1 ζ0 η0 φ_4(0) φ_ω(0) Γ0 = Fefermann Schutte Ordinal ψ(Ω^Ω^2) = φ(1,0,0,0) = Ackermann Ordinal ψ(ε_{Ω+1}) = Bachmann Howard Ordinal ψ(ε_{Ω_{ω}+1}}) = Takeuti Feferman Buchholz's Ordinal ψ(Ω_{ω+1}) = an ordinal greater than Takeuti Feferman Buchholz's Ordinal ψ(ψ_Ι(0)) = ψ(Ω_Ω_Ω_Ω_Ω_...) = ψ(Λ) = Extended Buchholz's Ordinal ω₁ᶜᴷ = Church Kleene Ordinal ω_1 = Ω = ℵ_1 = First Uncountable Ordinal Omega One / Aleph One Ω_Ω_Ω_Ω_Ω_.... = ψ_Ι(0) = ℵ_ℵ_ℵ_... = ω_ω_ω_... = Λ = Omega Fixed Point / First Aleph Fixed Point Gamma Cardinal Theta Cardinal I = Inacessible Cardinal M = Mahlo Cardinal K = Weakly Compact Cardinal Ξ = Π₀²-indescribable cardinal Π = Indescribable Cardinal i = Rank-into Rank Cardinal 0=1 = Zero equal one cardinal \color{red}{\Omega} = ABSOLUTE INFINITY!!! The standard names follow this pattern (short scale): - 10^6: million - 10^9: billion - 10^12: trillion - 10^15: quadrillion - 10^33: decillion - 10^36: undecillion - 10^63: vigintillion - 10^66: unvigintillion - 10^303: centillion - 10^306: uncentillion - 10^309: duocentillion - 10^312: trescentillion Examples: - Input: "1e6" -> "1 million" - Input: "1.5e7" -> "15 million" - Input: "2e9" -> "2 billion" - Input: "3.45e12" -> "3.45 trillion" - Input: "2^20" -> "1.049 million" - Input: "10^304" -> "10 centillion" - Input: "1 023" -> "1 023 (using spaces)"). **Standard Abbreviations:** - 10^6: M (Million) - 10^9: B (Billion) - 10^12: T (Trillion) - 10^15: Qa or q (Quadrillion) - 10^18: Qi or Q (Quintillion) - 10^21: Sx or s (Sextillion) - 10^24: Sp or S (Septillion) - 10^27: Oc or O (Octillion) - 10^30: No or N (Nonillion) - 10^33: Dc or D (Decillion) - 10^36: UDc or UD (Undecillion) - 10^39: DDc or DD (Duodecillion) - 10^42: TDc or TD (Tredecillion) - 10^45: qDc or qD (Quattourdecillion) - 10^48: QDc or QD (Quinquadecillion) - 10^63: Vg or V (Vigintillion) - 10^66: UV or UVg (Unvigintillion) - 10^303: Ce or C (Centillion) - 10^603: Duc or Du (Ducentillion) - 10^903: Tc or Te (Trecentillion/Trucentillion) - 10^3 003: MI (Millillion/Millinillion) **Beyond Millillion (Tier System based on N value where E = 3N + 3) Start of Tier 2:** **Millillion-based (N in thousands):** - N=1000 (millillion/millinillion): MI - N=1001 (milli-unillion): MI-U - N=1002 (milli-duillion): MI-D - N=1010 (milli-decillion): MI-De - N=1100 (milli-centillion): MI-Ce - N=2000 (dumillillion): DMI - N=3000 (trimillillion): TMI - N=10 000 (decimillillion): DeMI - N=20 000 (vigintimillillion): VgMI - N=100 000 (centimillillion): CeMI **Myrillion-based (alternative naming):** - N=10 000 (myrillion): MY - N=11 000 (myrimillillion): MY-MI - N=20 000 (dumyrillion): DMY - N=100 000 (decimyrillion): DeMY **Higher Tier Prefixes (based on N):** - N=10^6 (micrillion): MC - N=10^6+1 (micro-unillion): MC-U - N=10^6+1000 (micro-millillion): MC-MI - N=2*10^6 (dumicrillion): DMC - N=10^9 (nanillion): NA - N=2*10^9 (dunanillion): DNA - N=10^12 (picillion): PI - N=10^15 (femtillion): FM - N=10^18 (attillion): AT - N=10^21 (zeptillion): ZP - N=10^24 (yoctillion): YT - N=10^27 (xonillion/rontillion): Xn / Rn - N=10^30 (vecillion/quectillion): Vc / Qu - N=10^33 (mecillion): Me(c) - N=10^36 (duecillion): Du(c) - N=10^39 (trecillion): Tre(c) - N=10^42 (tetrecillion): Te(c) - N=10^45 (pentecillion): Pe(c) - N=10^48 (hexecillion): He(c) - N=10^51 (heptecillion): Hp(c) - N=10^54 (octecillion): Ot(c) - N=10^57 (ennecillion): En(c) - N=10^60 (icosillion): Is(o) - N=10^90 (triacontillion): TrC - N=10^120 (tetracontillion): TeC - N=10^150 (pentacontillion): PeC - N=10^180 (hexacontillion): HeC - N=10^210 (heptacontillion): HpC - N=10^240 (octacontillion): OtC - N=10^270 (enneacontillion): EnC - N=10^300 (hectillion): Hec - N=10^600 (dohectillion): DHc - N=10^900 (triahectillion): TrH - N=10^2997 (enneenneconteenneahectillion): EnEnCEnH. The tier-2 limit and tier-3 starts at: - N=10^3000 (killillion): Kl How to express, calculate equate, etc. it. (e.g. \begin{array}{rl} 3 \uparrow \uparrow \uparrow 3 &= 3 \uparrow \uparrow 7625597484987 \\ &\approx 10 \uparrow \uparrow 7625597484986.041 \approx 10\uparrow \uparrow (7.6256 \times 10^{12}) \\ &\approx 10 \uparrow \uparrow \uparrow 2.31076 \\ E1.11\#2\#2 &= E1.11\#(E1.11\#2) \\ &\approx 10\uparrow\uparrow(10\uparrow10\uparrow1.11) \\ &\approx 10\uparrow \uparrow (7.6295 \times 10^{12}) \end{array}) f_0(n) &=& n + 1 \\ f_1(n) &=& f_0^n(n) = ( \cdots ((n + 1) + 1) + \cdots + 1) = n + n = 2n \\ f_2(n) &=& f_1^n(n) = 2(2(\ldots 2(2n))) = 2^n n > 2^n \\ f_3(n) &>& 2\uparrow(2\uparrow(\ldots 2\uparrow(2\uparrow n))) > 2\uparrow\uparrow n\\ f_4(n) &>& 2\uparrow\uparrow(2\uparrow\uparrow(\ldots 2\uparrow\uparrow n)) > 2\uparrow\uparrow\uparrow n \\ f_m(n) &>& 2\uparrow^{m-1} n \\ f_\omega(n) &=& f_n(n) > 2\uparrow^{n-1} n > \text{Ack}(n) \\ f_{\omega+1}(n) &>& \lbrace n,n,1,2 \rbrace \\ f_{\omega+2}(n) &>& \lbrace n,n,2,2 \rbrace \\ f_{\omega+m}(n) &>& \lbrace n,n,m,2 \rbrace \\ f_{\omega\times 2}(n) &>& \lbrace n,n,n,2 \rbrace \\ f_{\omega\times 3}(n) &>& \lbrace n,n,n,3 \rbrace \\ f_{\omega\times m}(n) &>& \lbrace n,n,n,m \rbrace \\ f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace \\ f_{\omega^m}(n) &>& \lbrace n,m+2 [2] 2 \rbrace \\ f_{\omega^{\omega}}(n) &>& \lbrace n,n+2 [2] 2 \rbrace > \lbrace n,n [2] 2 \rbrace \\ f_{\omega^{\omega}+1}(n) &>& \lbrace n,n,2 [2] 2 \rbrace \\ f_{\omega^{\omega}+2}(n) &>& \lbrace n,n,3 [2] 2 \rbrace \\ f_{\omega^{\omega}+m}(n) &>& \lbrace n,n,m+1 [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega}(n) &>& \lbrace n,n,n+1 [2] 2 \rbrace > \lbrace n,n,n [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega+1}(n) &>& \lbrace n,n,1,2 [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega2}(n) &>& \lbrace n,n,n,2 [2] 2 \rbrace \\ f_{\omega^{\omega}+\omega^2}(n) &>& \lbrace n,n,n,n [2] 2 \rbrace \\ f_{{\omega^{\omega}}2}(n) &>& \lbrace n,n [2] 3 \rbrace \\ f_{{\omega^{\omega}}3}(n) &>& \lbrace n,n [2] 4 \rbrace \\ f_{{\omega^{\omega}}m}(n) &>& \lbrace n,n [2] m+1 \rbrace \\ f_{\omega^{\omega+1}}(n) &>& \lbrace n,n [2] n+1 \rbrace > \lbrace n,n [2] n \rbrace \\ f_{\omega^{\omega+2}}(n) &>& \lbrace n,n [2] n,n \rbrace \\ f_{\omega^{\omega+3}}(n) &>& \lbrace n,n,n [2] n,n,n \rbrace \\ f_{\omega^{\omega+m}}(n) &>& \lbrace n,m [2] 1 [2] 2 \rbrace \\ f_{\omega^{\omega2}}(n) &>& \lbrace n,n [2] 1 [2] 2 \rbrace > \lbrace n,2 [3] 2 \rbrace \\ f_{\omega^{\omega3}}(n) &>& \lbrace n,n [2] 1 [2] 1 [2] 2 \rbrace > \lbrace n,3 [3] 2 \rbrace \\ f_{\omega^{\omega m}}(n) &>& \lbrace n,m [3] 2 \rbrace \\ f_{\omega^{\omega^2}}(n) &>& \lbrace n,n [3] 2 \rbrace \\ f_{\omega^{\omega^3}}(n) &>& \lbrace n,n [4] 2 \rbrace \\ f_{\omega^{\omega^m}}(n) &>& \lbrace n,n [m+1] 2 \rbrace \\ f_{\omega^{\omega^\omega}}(n) &>& \lbrace n,n [n+1] 2 \rbrace > \lbrace n,n [1,2] 2 \rbrace \\ f_{^4{\omega}}(n) &>& \lbrace n,n [1 [2] 2] 2 \rbrace \\ f_{^5{\omega}}(n) &>& \lbrace n,n [1 [1,2] 2] 2 \rbrace \\ f_{^6{\omega}}(n) &>& \lbrace n,n [1 [1 [2] 2] 2] 2 \rbrace \\ f_{\varepsilon_0}(n) &>& \lbrace n,n [1 / 2] 2 \rbrace \\ f_{\varepsilon_02}(n) &>& \lbrace n,n [1 / 2] 3 \rbrace \\ f_{\varepsilon_03}(n) &>& \lbrace n,n [1 / 2] 4 \rbrace \\ f_{\varepsilon_0m}(n) &>& \lbrace n,n [1 / 2] m+1 \rbrace \\ f_{\varepsilon_0\omega}(n) &>& \lbrace n,n [1 / 2] n+1 \rbrace \\ f_{\varepsilon_0{\omega^{\omega}}}(n) &>& \lbrace n,n [1 / 2] 1 [2] 2 \rbrace \\ f_{\varepsilon_0{\omega^{\omega^{\omega}}}}(n) &>& \lbrace n,n [1 / 2] 1 [1,2] 2 \rbrace \\ f_{\varepsilon_0{\omega^{\omega^{\omega^{\omega}}}}}(n) &>& \lbrace n,n [1 / 2] 1 [1 [2] 2] 2 \rbrace \\ f_{\varepsilon_0^2}(n) &>& \lbrace n,n [1 / 2] 1 [1 / 2] 2 \rbrace \\ f_{\varepsilon_0^3}(n) &>& \lbrace n,n [1 / 2] 1 [1 / 2] 1 [1 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\omega}}(n) &>& \lbrace n,n [2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\omega^{\omega}}}(n) &>& \lbrace n,n [1,2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\omega^{\omega^{\omega}}}}(n) &>& \lbrace n,n [1 [2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\varepsilon_0}}(n) &>& \lbrace n,n [1 [1 / 2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(n) &>& \lbrace n,n [1 [1 / 2] 1 [1 / 2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(n) &>& \lbrace n,n [1 [1 [1 / 2] 2 / 2] 2 / 2] 2 \rbrace \\ f_{\varepsilon_1}(n) &>& \lbrace n,n [1 / 3] 2 \rbrace \\ f_{\varepsilon_2}(n) &>& \lbrace n,n [1 / 4] 2 \rbrace \\ f_{\varepsilon_{\omega}}(n) &>& \lbrace n,n [1 / 1,2] 2 \rbrace \\ f_{\varepsilon_{\omega^2}}(n) &>& \lbrace n,n [1 / 1,1,2] 2 \rbrace \\ f_{\varepsilon_{\omega^{\omega}}}(n) &>& \lbrace n,n [1 / 1 [2] 2] 2 \rbrace \\ f_{\varepsilon_{\omega^{\omega^{\omega}}}}(n) &>& \lbrace n,n [1 / 1 [1,2] 2] 2 \rbrace \\ f_{\varepsilon_{\varepsilon_0}}(n) &>& \lbrace n,n [1 / 1 [1 / 2] 2] 2 \rbrace \\ f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}(n) &>& \lbrace n,n [1 / 1 [1 / 1 [1 / 2] 2] 2] 2 \rbrace \\ f_{\zeta_0}(n) &>& \lbrace n,n [1 / 1 / 2] 2 \rbrace \\ f_{\zeta_0^{\zeta_0}}(n) &>& \lbrace n,n [1 [1 / 1 / 2] 2 / 1 / 2] 2 \rbrace \\ f_{\varepsilon_{\zeta_0+1}}(n) &>& \lbrace n,n [1 / 2 / 2] 2 \rbrace \\ f_{\varepsilon_{\zeta_0+2}}(n) &>& \lbrace n,n [1 / 3 / 2] 2 \rbrace \\ f_{\varepsilon_{\varepsilon_{\zeta_0+1}}}(n) &>& \lbrace n,n [1 / 1 [1 / 2 / 2] 2 / 2]2 \rbrace \\ f_{\zeta_1}(n) &>& \lbrace n,n [1 / 1 / 3] 2 \rbrace \\ f_{\zeta_2}(n) &>& \lbrace n,n [1 / 1 / 4] 2 \rbrace \\ f_{\zeta_{\zeta_0}}(n) &>& \lbrace n,n [1 / 1 / 1 [1 / 1 / 2] 2] 2 \rbrace \\ f_{\eta_0}(n) &>& \lbrace n,n [1 / 1 / 1 / 2] 2 \rbrace \\ f_{\varphi(4,0)}(n) &>& \lbrace n,n [1 / 1 / 1 / 1 / 2] 2 \rbrace \\ f_{\varphi(\omega,0)}(n) &>& \lbrace n,n [1 [2 \sim 2] 2] 2 \rbrace \\ f_{\varphi(\varphi(\omega,0),0)}(n) &>& \lbrace n,n [1 [1 [1 [2 \sim 2] 2] 2 \sim 2] 2] 2 \rbrace \\ f_{\Gamma_0}(n) &>& \lbrace n,n [1 [1 / 2 \sim 2] 2] 2 \rbrace \\ f_{\varphi(1,0,0,0)}(n) &>& \lbrace n,n [1 [1 / 3 \sim 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega^{\omega})}(n) &>& \lbrace n,n [1 [1 / 1, 2 \sim 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega^{\Omega})}(n) &>& \lbrace n,n [1 [1 / 1 / 2 \sim 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega^{\Omega^{\Omega}})}(n) &>& \lbrace n,n [1 [1 [1 / 2 \sim 2] 2 \sim 2] 2] 2 \rbrace \\ f_{\vartheta(\vartheta_1(1))}(n) &>& \lbrace n,n [1 [1 \sim 3] 2] 2 \rbrace \\ f_{\vartheta(\vartheta_1(2))}(n) &>& \lbrace n,n [1 [1 \sim 1 \sim 2] 2] 2 \rbrace \\ f_{\vartheta(\vartheta_1(\omega))}(n) &>& \lbrace n,n [1 [1 [2 /_3 2] 2] 2] 2 \rbrace \\ f_{\vartheta(\vartheta_1(\Omega))}(n) &>& \lbrace n,n [1 [1 [1 / 2 /_3 2] 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_2)}(n) &>& \lbrace n,n [1 [1 [1 \sim 2 /_3 2] 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_3)}(n) &>& \lbrace n,n [1 [1 [1 [1 /_3 2 /_4 2] 2] 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\omega})}(n) &>& \lbrace n,n [1 [2 /_{1,2} 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\varepsilon_0})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 / 2] 2} 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\Gamma_0})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 / 2 \sim 2] 2} 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\vartheta(\Omega_2)})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 [1 [1 \sim2 /_3 2] 2] 2] 2} 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\vartheta(\Omega_3)})}(n) &>& \lbrace n,n [1 [2/_{1 [1 [1 [1 [1 /_3 2/_4 2] 2] 2] 2] 2} 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\vartheta(\Omega_{\omega})})}(n) &>& \lbrace n,n [1 [2 /_{1 [1 [2 /_{1,2} 2] 2] 2} 2] 2] 2 \rbrace \\ f_{\vartheta(\Omega_{\vartheta(\Omega_{\vartheta(\Omega_{\omega})})})}(n) &>& \lbrace n,n [1 [2 /_{1 [2 /_{1 [1 [2 /_{1,2} 2] 2] 2} 2] 2} 2] 2] 2 \rbrace \end{eqnarray*}. S = {0,1,ω,Ω} C(0) = {Finite AME 0,1,ω,Ω) ψ(x) = smallest ordinal not a member of C(x) ω+1, ωx2, ω^ω, .....(infinite),..... = ε_0 = ψ(0) ψ(1) = ε_1 ψ(ω) = ε_ω ψ(ε_ε_ε_...) = ψ(ψ(ψ(...(...)))))....))) = ψ(ζ_0) = ψ(Ω) = ζ_0 ψ(Ω+1) = ε_{ζ_{0}+1}} ψ(Ω+2) = ε_{ζ_{0}+2}} ψ(Ω+a) = ε_{ζ_{0}+a}} ψ(Ω2) = ζ_1 ψ(Ω2) = ζ_2 ψ(Ωω) = ζ_ω ψ(Ω^2)= η_0 ψ(Ω^3)= φ_{4}(0) = φ(4,0) ψ(Ω^ω)= φ_{ω}(0) = φ(ω,0) ψ(Ω^Ω) = Γ_0 = φ(1,0,0) ψ(Ω^{Ω}a) = Γ_{a-1} ψ(Ω^{Ω+1}) = φ(1,1,0) ψ(Ω^{Ω+a}) = φ(1,a,0) ψ(Ω^{Ωa}) = φ(a,0,0) ψ(Ω^Ω^2) = φ(1,0,0,0) ψ(Ω^Ω^3) = φ(1,0,0,0,0) ψ(Ω^Ω^ω) = φ(1,0,0,0,0,0,0,......) (ω times)= SVO = φ(1 @^ω) ψ(Ω^Ω^Ω) = φ(1,0,0,0,0,0,0,......) (Ω times) = LVO = φ(1 @^{1,0}) We can continue exponentiating Ω's and more (e.g. φ(1 @^{1,0}, φ(1 @^{1 @^{1,0}}, etc.) like this, but only a finite number of times. If we do it an infinite number of times, then we'll reach our next ordinal which is.... ψ(Ω^Ω^Ω^...) = ψ(Ω↑↑ω) = ψ(ε_{Ω+1}) = ψ(ψ_1(0)) = ψ_1(0) = ψ(Ω_2) (in Buchholz's function, extended from Googology Wiki) = BHO \underbrace{C(C(\cdots(C(C(\Omega_{n}2,0),0),\cdots ),0)}_{n\text{ C's}} = Taranovsky's function Use this format: "1000" (no spaces) (as 10^3), "10 000" (as 10^4), etc. Example of a respect to an unspecified (and perhaps ill-defined) OCF \psi Ordinal List: 1, 2, 3, 4, 5, "$\\omega$", "$\\omega+1$", "$\\omega+2$", "$\\omega+3$", "$\\omega+4$", "$\\omega+5$", "$\\omega 2$", "$\\omega 3$", "$\\omega 4$", "$\\omega 5$", "$\\omega^2$", "$\\omega^3$", "$\\omega^4$", "$\\omega^5$", "$\\omega^\\omega$", "$\\omega^{\\omega^{\\omega}}$", "$\\omega^{\\omega^{\\omega^{\\omega}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\omega}}}}}$", "$\\varepsilon_0$", "$\\varepsilon_0+1$", "$\\varepsilon_0+2$", "$\\varepsilon_0+3$", "$\\varepsilon_0+4$", "$\\varepsilon_0+5$", "$\\varepsilon_0 2$", "$\\varepsilon_0 3$", "$\\varepsilon_0 4$", "$\\varepsilon_0 5$", "$\\omega^{\\varepsilon_0+1}$", "$\\omega^{\\varepsilon_0+1}+1$", "$\\omega^{\\varepsilon_0+1}2$", "$\\omega^{\\varepsilon_0+1}3$", "$\\omega^{\\varepsilon_0+2}$", "$\\omega^{\\varepsilon_0+3}$", "$\\omega^{\\varepsilon_0+4}$", "$\\omega^{\\varepsilon_0+5}$", "$\\omega^{\\varepsilon_0+5}$", "$\\omega^{\\varepsilon_0 2}$", "$\\omega^{\\varepsilon_0 3}$", "$\\omega^{\\varepsilon_0 4}$", "$\\omega^{\\varepsilon_0 5}$", "$\\omega^{\\omega^{\\varepsilon_0+1}}$", "$\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_0+1}}}}}}$", "$\\varepsilon_1$", "$\\varepsilon_1+1$", "$\\varepsilon_1+2$", "$\\varepsilon_1+3$", "$\\varepsilon_1+4$", "$\\varepsilon_1+5$", "$\\varepsilon_1+\\omega$", "$\\varepsilon_1+\\omega 2$", "$\\varepsilon_1+\\omega 3$", "$\\varepsilon_1+\\omega 4$", "$\\varepsilon_1+\\omega^{\\omega}$", "$\\varepsilon_1+\\omega^{\\omega^{\\omega}}$", "$\\varepsilon_1+\\varepsilon_0$", "$\\varepsilon_1+\\omega^{\\varepsilon_0}$", "$\\varepsilon_1+\\omega^{\\omega^{\\varepsilon_0}}$", "$\\varepsilon_1+\\omega^{\\omega^{\\omega^{\\varepsilon_0}}}$", "$\\varepsilon_1 2$", "$\\varepsilon_1 3$", "$\\varepsilon_1 4$", "$\\varepsilon_1 5$", "$\\omega^{\\varepsilon_1+1}$", "$\\omega^{\\varepsilon_1+2}$", "$\\omega^{\\varepsilon_1+3}$", "$\\omega^{\\varepsilon_1+4}$", "$\\omega^{\\varepsilon_1+5}$", "$\\omega^{\\varepsilon_1 2}$", "$\\omega^{\\varepsilon_1 3}$", "$\\omega^{\\varepsilon_1 4}$", "$\\omega^{\\varepsilon_1 5}$", "$\\omega^{\\omega^{\\varepsilon_1+1}}$", "$\\omega^{\\omega^{\\omega^{\\varepsilon_1+1}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_1+1}}}}$", "$\\omega^{\\omega^{\\omega^{\\omega^{\\omega^{\\varepsilon_1+1}}}}}$", "$\\varepsilon_2$", "$\\varepsilon_3$", "$\\varepsilon_4$", "$\\varepsilon_5$", "$\\varepsilon_6$", "$\\varepsilon_\\omega$", "$\\varepsilon_{\\omega+1}$", "$\\varepsilon_{\\omega+2}$", "$\\varepsilon_{\\omega+3}$", "$\\varepsilon_{\\omega+4}$", "$\\varepsilon_{\\omega 2}$", "$\\varepsilon_{\\omega 3}$", "$\\varepsilon_{\\omega 4}$", "$\\varepsilon_{\\omega^{\\omega^\\omega}}$", "$\\varepsilon_{\\varepsilon_0}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}}}$", "$\\zeta_0$", "$\\varepsilon_{\\zeta_0}$", "$\\varepsilon_{\\varepsilon_{\\zeta_0}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\zeta_0}}}$", "$\\varepsilon_{\\varepsilon_{\\varepsilon_{\\varepsilon_{\\zeta_0} }}}$", "$\\zeta_1$", "$\\zeta_2$", "$\\zeta_3$", "$\\zeta_4$", "$\\zeta_5$", "$\\zeta_\\omega$", "$\\zeta_{\\omega^2}$", "$\\zeta_{\\omega^3}$", "$\\zeta_{\\omega^4}$", "$\\zeta_{\\omega^\\omega}$", "$\\zeta_{\\omega^{\\omega^\\omega}}$", "$\\zeta_{\\varepsilon_{0}}$", "$\\zeta_{\\varepsilon_{1}}$", "$\\zeta_{\\varepsilon_{2}}$", "$\\zeta_{\\varepsilon_{3}}$", "$\\zeta_{\\varepsilon_{4}}$", "$\\zeta_{\\varepsilon_{\\varepsilon_{0}}}$", "$\\zeta_{\\varepsilon_{\\varepsilon_{\\varepsilon_{0}}}}$", "$\\zeta_{\\zeta_{0}}$", "$\\zeta_{\\zeta_{\\zeta_{0}}}$", "$\\zeta_{\\zeta_{\\zeta_{\\zeta_{0}}}}$", "$\\varphi_{3}(0)$", "$\\varphi_{3}(1)$", "$\\varphi_{3}(2)$", "$\\varphi_{3}(3)$", "$\\varphi_{3}(4)$", "$\\varphi_{3}(5)$", "$\\varphi_{3}(\\omega)$", "$\\varphi_{3}(\\varepsilon_0)$", "$\\varphi_{3}(\\zeta_0)$", "$\\varphi_{3}(\\varphi_{3}(0))$", "$\\varphi_{3}(\\varphi_{3}(\\varphi_{3}(0)))$", "$\\varphi_{3}(\\varphi_{3}(\\varphi_{3}(\\varphi_{3}(0))))$", "$\\varphi_{4}(0)$", "$\\varphi_{4}(1)$", "$\\varphi_{4}(2)$", "$\\varphi_{4}(3)$", "$\\varphi_{4}(\\omega)$", "$\\varphi_{4}(\\varepsilon_{0})$", "$\\varphi_{4}(\\zeta_{0})$", "$\\varphi_{4}(\\varphi_{3}(0))$", "$\\varphi_{4}(\\varphi_{4}(0))$", "$\\varphi_{5}(0)$", "$\\varphi_{6}(0)$", "$\\varphi_{7}(0)$", "$\\varphi_{8}(0)$", "$\\varphi_{\\omega}(0)$", "$\\varphi_{\\varepsilon_{0}}(0)$", "$\\varphi_{\\zeta_{0}}(0)$", "$\\varphi_{\\varphi_{3}(0)}(0)$", "$\\varphi_{\\varphi_{4}(0)}(0)$", "$\\varphi_{\\varphi_{5}(0)}(0)$", "$\\varphi_{\\varphi_{6}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)}(0)}(0)$", "$\\varphi_{\\varphi_{\\varphi_{\\varphi_{\\varphi_{\\varphi_{1}(0)}(0)}(0)}(0)}(0)}(0)$", "$\\varphi(1,0,0)$", "$\\varphi(1,0,1)$", "$\\varphi(1,0,2)$", "$\\varphi(1,0,3)$", "$\\varphi(1,0,4)$", "$\\varphi(1,0,5)$", "$\\varphi(1,0,\\omega)$", "$\\varphi(1,0,\\varepsilon_{0})$", "$\\varphi(1,0,\\varphi_{2}(0))$", "$\\varphi(1,0,\\varphi_{3}(0))$", "$\\varphi(1,0,\\varphi_{4}(0))$", "$\\varphi(1,0,\\varphi_{5}(0))$", "$\\varphi(1,0,\\varphi_{\\omega}(0))$", "$\\varphi(1,0,\\varphi_{\\varepsilon_{0}}(0))$", "$\\varphi(1,0,\\varphi_{\\zeta_{0}}(0))$", "$\\varphi(1,0,\\varphi(1,0,0))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0)))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0)))))$", "$\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))))))$", "$\\varphi(1,1,0)$", "$\\varphi(1,1,1)$", "$\\varphi(1,1,2)$", "$\\varphi(1,1,3)$", "$\\varphi(1,1,4)$", "$\\varphi(1,1,5)$", "$\\varphi(1,1,\\omega)$", "$\\varphi(1,1,\\varepsilon_{0})$", "$\\varphi(1,1,\\zeta_{0})$", "$\\varphi(1,2,0)$", "$\\varphi(1,2,1)$", "$\\varphi(1,2,2)$", "$\\varphi(1,2,3)$", "$\\varphi(1,2,4)$", "$\\varphi(1,2,5)$", "$\\varphi(1,3,0)$", "$\\varphi(1,4,0)$", "$\\varphi(1,5,0)$", "$\\varphi(1,6,0)$", "$\\varphi(1,7,0)$", "$\\varphi(1,8,0)$", "$\\varphi(1,\\omega,0)$", "$\\varphi(1,\\varepsilon_{0},0)$", "$\\varphi(1,\\zeta_{0},0)$", "$\\varphi(1,\\varphi_{3}(0),0)$", "$\\varphi(1,\\varphi_{\\omega}(0),0)$", "$\\varphi(1,\\varphi_{\\varepsilon_{0}}(0),0)$", "$\\varphi(1,\\varphi_{\\zeta_{0}}(0),0)$", "$\\varphi(1,\\varphi(1,0,0),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,0)),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0)))),0)$", "$\\varphi(1,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,\\varphi(1,0,0))))),0)$", "$\\varphi(2,0,0)$", "$\\varphi(2,0,1)$", "$\\varphi(2,0,2)$", "$\\varphi(2,0,3)$", "$\\varphi(2,0,4)$", "$\\varphi(2,0,5)$", "$\\varphi(2,1,0)$", "$\\varphi(2,2,0)$", "$\\varphi(2,3,0)$", "$\\varphi(2,4,0)$", "$\\varphi(2,5,0)$", "$\\varphi(2,6,0)$", "$\\varphi(3,0,0)$", "$\\varphi(4,0,0)$", "$\\varphi(5,0,0)$", "$\\varphi(6,0,0)$", "$\\varphi(7,0,0)$", "$\\varphi(8,0,0)$", "$\\varphi(1,0,0,0)$", "$\\varphi(1,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0,0,0)$", "$\\varphi(1,0,0,0,0,0,0,0,0,0)$", "$\\psi(\\Omega^{\\Omega^\\omega})$", "$\\psi(\\Omega^{\\Omega^\\omega+1})$", "$\\psi(\\Omega^{\\Omega^\\omega+2})$", "$\\psi(\\Omega^{\\Omega^\\omega+3})$", "$\\psi(\\Omega^{\\Omega^\\omega 2})$", "$\\psi(\\Omega^{\\Omega^\\omega 3})$", "$\\psi(\\Omega^{\\Omega^\\omega 4})$", "$\\psi(\\Omega^{\\Omega^\\omega 5})$", "$\\psi(\\Omega^{\\Omega^\\Omega})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+1}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+2}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+3}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+4}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega+5}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 2}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 3}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 4}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega 5}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{2}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{3}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{4}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{5}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\omega}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}}}})$", "$\\psi(\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega^{\\Omega}}}}}}})$", "$\\psi(\\Omega_2)$", "$\\psi(\\Omega_2+1)$", "$\\psi(\\Omega_2+2)$", "$\\psi(\\Omega_2+3)$", "$\\psi(\\Omega_2+4)$", "$\\psi(\\Omega_2+5)$", "$\\psi(\\Omega_2+\\omega)$", "$\\psi(\\Omega_2+\\Omega)$", "$\\psi(\\Omega_2 2)$", "$\\psi(\\Omega_2 3)$", "$\\psi(\\Omega_2 4)$", "$\\psi(\\Omega_2 5)$", "$\\psi(\\Omega_2 \\omega)$", "$\\psi(\\Omega_2 \\Omega)$", "$\\psi(\\Omega_2^2)$", "$\\psi(\\Omega_2^3)$", "$\\psi(\\Omega_2^4)$", "$\\psi(\\Omega_2^5)$", "$\\psi(\\Omega_2^{\\Omega_2})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}}}})$", "$\\psi(\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2^{\\Omega_2}}}}}})$", "$\\psi(\\Omega_3)$", "$\\psi(\\Omega_3^{\\Omega_3})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3}})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3}}})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3}}}})$", "$\\psi(\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3^{\\Omega_3}}}}})$", "$\\psi(\\Omega_4)$", "$\\psi(\\Omega_5)$", "$\\psi(\\Omega_6)$", "$\\psi(\\Omega_7)$", "$\\psi(\\Omega_8)$", "$\\psi(\\Omega_9)$", "$\\psi(\\Omega_\\omega)$", "$\\psi(\\Omega_\\Omega)$", "$\\psi(\\Omega_{\\Omega_{\\omega}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega}}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega}}}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega}}}}})$", "$\\psi(\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega_{\\Omega}}}}}})$", "$\\psi(\\psi_I(0))$", "$\\psi(\\psi_I(0)+1)$", "$\\psi(\\psi_I(0)+2)$", "$\\psi(\\psi_I(0)+3)$", "$\\psi(\\psi_I(0)+4)$", "$\\psi(\\psi_I(0)+5)$", "$\\psi(\\psi_I(0)2)$", "$\\psi(\\psi_I(0)3)$", "$\\psi(\\psi_I(0)4)$", "$\\psi(\\psi_I(0)5)$", "$\\psi(\\psi_I(0)^{\\psi_I(0)})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}}})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}}}})$", "$\\psi(\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)^{\\psi_I(0)}}}}})$", "$\\psi(\\psi_I(1))$", "$\\psi(\\psi_I(1)^{\\psi_I(1)})$", "$\\psi(\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)}})$", "$\\psi(\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)}}})$", "$\\psi(\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)^{\\psi_I(1)}}}})$", "$\\psi(\\psi_I(3))$", "$\\psi(\\psi_I(4))$", "$\\psi(\\psi_I(5))$", "$\\psi(\\psi_I(6))$", "$\\psi(\\psi_I(\\omega))$", "$\\psi(\\psi_I(\\Omega))$", "$\\psi(\\psi_I(\\psi_I(0)))$", "$\\psi(\\psi_I(\\psi_I(\\psi_I(0))))$", "$\\psi(\\psi_I(\\psi_I(\\psi_I(\\psi_I(0)))))$", "$\\psi(\\psi_I(\\psi_I(\\psi_I(\\psi_I(\\psi_I(0))))))$", "$\\psi(I)$", "$\\psi(I+1)$", "$\\psi(I+2)$", "$\\psi(I+3)$", "$\\psi(I+4)$", "$\\psi(I+5)$", "$\\psi(I+\\omega)$", "$\\psi(I+\\Omega)$", "$\\psi(I+\\Omega_2)$", "$\\psi(I+\\Omega_3)$", "$\\psi(I+\\Omega_4)$", "$\\psi(I+\\Omega_5)$", "$\\psi(I+\\Omega_\\omega)$", "$\\psi(I+\\Omega_\\Omega)$", "$\\psi(I+\\Omega_{\\Omega_\\Omega})$", "$\\psi(I+\\psi_I(0))$", "$\\psi(I+\\psi_I(1))$", "$\\psi(I+\\psi_I(2))$", "$\\psi(I+\\psi_I(3))$", "$\\psi(I+\\psi_I(4))$", "$\\psi(I+\\psi_I(5))$", "$\\psi(I+\\psi_I(\\omega))$", "$\\psi(I+\\psi_I(\\Omega))$", "$\\psi(I+\\psi_I(\\psi_I(0)))$", "$\\psi(I+\\psi_I(\\psi_I(\\psi_I(0))))$", "$\\psi(I+\\psi_I(\\psi_I(\\psi_I(\\psi_I(0)))))$", "$\\psi(I2)$", "$\\psi(I3)$", "$\\psi(I4)$", "$\\psi(I5)$", "$\\psi(I6)$", "$\\psi(I\\omega)$", "$\\psi(I\\Omega)$", "$\\psi(I\\psi_I(0))$", "$\\psi(I\\psi_I(\\psi_I(0)))$", "$\\psi(I\\psi_I(\\psi_I(\\psi_I(0))))$", "$\\psi(I^2)$", "$\\psi(I^3)$", "$\\psi(I^4)$", "$\\psi(I^5)$", "$\\psi(I^6)$", "$\\psi(I^\\omega)$", "$\\psi(I^\\Omega)$", "$\\psi(I^{\\psi_I(0)})$", "$\\psi(I^{\\psi_I(1)})$", "$\\psi(I^{\\psi_I(2)})$", "$\\psi(I^{\\psi_I(3)})$", "$\\psi(I^{\\psi_I(4)})$", "$\\psi(I^{\\psi_I(\\omega)})$", "$\\psi(I^{\\psi_I(\\Omega)})$", "$\\psi(I^{\\psi(I)})$", "$\\psi(I^I)$", "$\\psi(I^{I^2})$", "$\\psi(I^{I^3})$", "$\\psi(I^{I^4})$", "$\\psi(I^{I^5})$", "$\\psi(I^{I^I})$", "$\\psi(I^{I^{I^I}})$", "$\\psi(I^{I^{I^{I^{I}}}})$", "$\\psi(I^{I^{I^{I^{I^{I}}}}})$", "$\\psi(I^{I^{I^{I^{I^{I^{I}}}}}})$", "$\\psi(\\psi_{I_2}(0))$", "$\\psi(\\psi_{I_2}(1))$", "$\\psi(\\psi_{I_2}(2))$", "$\\psi(\\psi_{I_2}(3))$", "$\\psi(\\psi_{I_2}(4))$", "$\\psi(\\psi_{I_2}(5))$", "$\\psi(I_2)$", "$\\psi(I_2+1)$", "$\\psi(I_2+2)$", "$\\psi(I_2+3)$", "$\\psi(I_2+4)$", "$\\psi(I_2+5)$", "$\\psi(I_2 2)$", "$\\psi(I_2 3)$", "$\\psi(I_2 4)$", "$\\psi(I_2 5)$", "$\\psi(I_2^{I_2})$", "$\\psi(I_2^{I_2^{I_2}})$", "$\\psi(I_2^{I_2^{I_2^{I_2}}})$", "$\\psi(I_2^{I_2^{I_2^{I_2^{I_2}}}})$", "$\\psi(I_3)$", "$\\psi(I_4)$", "$\\psi(I_5)$", "$\\psi(I_6)$", "$\\psi(I_7)$", "$\\psi(I_{\\omega+1})$", "$\\psi(I_{\\omega+2})$", "$\\psi(I_{\\omega+3})$", "$\\psi(I_{\\omega+4})$", "$\\psi(I_{\\omega 2})$", "$\\psi(I_{\\omega 3})$", "$\\psi(I_{\\omega 4})$", "$\\psi(I_{\\omega 5})$", "$\\psi(I_{\\Omega})$", "$\\psi(I_{\\psi_I(0)})$", "$\\psi(I_{\\psi(I)})$", "$\\psi(I_{\\psi(I^2)})$", "$\\psi(I_{\\psi(I^3)})$", "$\\psi(I_{\\psi(I^4)})$", "$\\psi(I_{I})$", "$\\psi(I_{I_{I}})$", "$\\psi(I_{I_{I_{I}}})$", "$\\psi(I_{I_{I_{I_{I}}}})$", "$\\psi(I_{I_{I_{I_{I_{I}}}}})$", "$\\psi(\\psi_{I(1,0)}(0))$", "$\\psi(\\psi_{I(1,0)}(1))$", "$\\psi(\\psi_{I(1,0)}(2))$", "$\\psi(\\psi_{I(1,0)}(3))$", "$\\psi(\\psi_{I(1,0)}(4))$", "$\\psi(\\psi_{I(1,0)}(5))$", "$\\psi(I(1,0))$", "$\\psi(I(1,0)+1)$", "$\\psi(I(1,0)+2)$", "$\\psi(I(1,0)+3)$", "$\\psi(I(1,0)+4)$", "$\\psi(I(1,0)+5)$", "$\\psi(I(1,0)2)$", "$\\psi(I(1,0)3)$", "$\\psi(I(1,0)4)$", "$\\psi(I(1,0)5)$", "$\\psi(I(1,0)^2)$", "$\\psi(I(1,0)^2 2)$", "$\\psi(I(1,0)^2 3)$", "$\\psi(I(1,0)^2 4)$", "$\\psi(I(1,0)^2 5)$", "$\\psi(I(1,0)^3)$", "$\\psi(I(1,0)^4)$", "$\\psi(I(1,0)^5)$", "$\\psi(I(1,0)^6)$", "$\\psi(I(1,0)^I)$", "$\\psi(I(1,0)^{I_2})$", "$\\psi(I(1,0)^{I_3})$", "$\\psi(I(1,0)^{I_4})$", "$\\psi(I(1,0)^{I_5})$", "$\\psi(I(1,0)^{I_I})$", "$\\psi(I(1,0)^{I_{I_{I}}})$", "$\\psi(I(1,0)^{I_{I_{I_{I}}}})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(0)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(1)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(2)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(3)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(4)})$", "$\\psi(I(1,0)^{\\psi_{I(1,0)}(5)})$", "$\\psi(I(1,0)^{I(1,0)})$", "$\\psi(I(1,0)^{I(1,0)^{I(1,0)}})$", "$\\psi(I(1,0)^{I(1,0)^{I(1,0)^{I(1,0)}}})$", "$\\psi(I(1,0)^{I(1,0)^{I(1,0)^{I(1,0)^{I(1,0)}}}})$", "$\\psi(I(1,1))$", "$\\psi(I(1,2))$", "$\\psi(I(1,3))$", "$\\psi(I(1,4))$", "$\\psi(I(1,5))$", "$\\psi(I(1,\\omega+1))$", "$\\psi(I(1,\\Omega))$", "$\\psi(I(1,\\psi_{I(1,0)}(0)))$", "$\\psi(I(1,\\psi(I(1,0))))$", "$\\psi(I(2,0))$", "$\\psi(I(2,1))$", "$\\psi(I(2,2))$", "$\\psi(I(2,3))$", "$\\psi(I(2,4))$", "$\\psi(I(2,5))$", "$\\psi(I(3,0))$", "$\\psi(I(4,0))$", "$\\psi(I(5,0))$", "$\\psi(I(6,0))$", "$\\psi(M^M)$", "$\\psi(M^M+1)$", "$\\psi(M^M+2)$", "$\\psi(M^M+3)$", "$\\psi(M^M+4)$", "$\\psi(M^M+5)$", "$\\psi(M^M+\\omega)$", "$\\psi(M^M+\\Omega)$", "$\\psi(M^M+\\psi_I(0))$", "$\\psi(M^M+\\psi_I(1))$", "$\\psi(M^M+\\psi_I(2))$", "$\\psi(M^M+\\psi_I(3))$", "$\\psi(M^M+M)$", "$\\psi(M^M+M^2)$", "$\\psi(M^M+M^3)$", "$\\psi(M^M+M^4)$", "$\\psi(M^M+M^5)$", "$\\psi(M^M2)$", "$\\psi(M^M3)$", "$\\psi(M^M4)$", "$\\psi(M^M5)$", "$\\psi(M^M6)$", "$\\psi(M^{M+1})$", "$\\psi(M^{M+2})$", "$\\psi(M^{M+3})$", "$\\psi(M^{M+4})$", "$\\psi(M^{M+5})$", "$\\psi(M^{M2})$", "$\\psi(M^{M3})$", "$\\psi(M^{M4})$", "$\\psi(M^{M5})$", "$\\psi(M^{M^2})$", "$\\psi(M^{M^2}+1)$", "$\\psi(M^{M^2}+2)$", "$\\psi(M^{M^2}+3)$", "$\\psi(M^{M^2}+4)$", "$\\psi(M^{M^2}+5)$", "$\\psi(M^{M^2}+\\omega)$", "$\\psi(M^{M^2}+\\Omega)$", "$\\psi(M^{M^2}+M)$", "$\\psi(M^{M^2}+M^2)$", "$\\psi(M^{M^2}+M^3)$", "$\\psi(M^{M^2}+M^4)$", "$\\psi(M^{M^2}+M^5)$", "$\\psi(M^{M^2}+M^M)$", "$\\psi(M^{M^2}2)$", "$\\psi(M^{M^2}3)$", "$\\psi(M^{M^2}4)$", "$\\psi(M^{M^2}5)$", "$\\psi(M^{M^2+1})$", "$\\psi(M^{M^2+2})$", "$\\psi(M^{M^2+4})$", "$\\psi(M^{M^2+5})$", "$\\psi(M^{M^3})$", "$\\psi(M^{M^4})$", "$\\psi(M^{M^5})$", "$\\psi(M^{M^6})$", "$\\psi(M^{M^M})$", "$\\psi(M^{M^{M^{M}}})$", "$\\psi(M^{M^{M^{M^{M}}}})$", "$\\psi(M^{M^{M^{M^{M^{M}}}}})$", "$\\psi(M^{M^{M^{M^{M^{M^{M}}}}}})$", "$\\psi(M^{M^{M^{M^{M^{M^{M^{M}}}}}}})$", "$\\psi(\\psi_{M_2}(0))$", "$\\psi(\\psi_{M_2}(1))$", "$\\psi(\\psi_{M_2}(2))$", "$\\psi(\\psi_{M_2}(3))$", "$\\psi(\\psi_{M_2}(4))$", "$\\psi(\\psi_{M_2}(5))$", "$\\psi(M_2)$", "$\\psi(M_2+1)$", "$\\psi(M_2+2)$", "$\\psi(M_2+3)$", "$\\psi(M_2+4)$", "$\\psi(M_2+5)$", "$\\psi(M_2+\\omega)$", "$\\psi(M_2+\\Omega)$", "$\\psi(M_2+M)$", "$\\psi(M_2 2)$", "$\\psi(M_2 3)$", "$\\psi(M_2 4)$", "$\\psi(M_2 5)$", "$\\psi(M_2 6)$", "$\\psi(M_2^2)$", "$\\psi(M_2^2+1)$", "$\\psi(M_2^2+2)$", "$\\psi(M_2^2+3)$", "$\\psi(M_2^2+4)$", "$\\psi(M_2^2+5)$", "$\\psi(M_2^2 2)$", "$\\psi(M_2^2 3)$", "$\\psi(M_2^2 4)$", "$\\psi(M_2^2 5)$", "$\\psi(M_2^3)$", "$\\psi(M_2^4)$", "$\\psi(M_2^5)$", "$\\psi(M_2^6)$", "$\\psi(M_2^M)$", "$\\psi(M_2^{M_2})$", "$\\psi(M_2^{M_2^{M_2}})$", "$\\psi(M_2^{M_2^{M_2^{M_2}}})$", "$\\psi(M_2^{M_2^{M_2^{M_2^{M_2}}}})$", "$\\psi(M_2^{M_2^{M_2^{M_2^{M_2^{M_2}}}}})$", "$\\psi(M_2^{M_2^{M_2^{M_2^{M_2^{M_2^{M_2}}}}}})$", "$\\psi(M_3)$", "$\\psi(M_4)$", "$\\psi(M_5)$", "$\\psi(M_6)$", "$\\psi(M_7)$", "$\\psi(M_8)$", "$\\psi(M_9)$", "$\\psi(M_{\\omega+1})$", "$\\psi(M_{\\omega+2})$", "$\\psi(M_{\\omega+3})$", "$\\psi(M_{\\omega+4})$", "$\\psi(M_{\\omega+5})$", "$\\psi(M_\\Omega)$", "$\\psi(M_{M})$", "$\\psi(M_{M_{2}})$", "$\\psi(M_{M_{3}})$", "$\\psi(M_{M_{4}})$", "$\\psi(M_{M_{5}})$", "$\\psi(M_{M_{M}})$", "$\\psi(M_{M_{M_{M}}})$", "$\\psi(M_{M_{M_{M_{M}}}})$", "$\\psi(M_{M_{M_{M_{M_{M}}}}})$", "$\\psi(M_{M_{M_{M_{M_{M_{M}}}}}})$", "$\\psi(M_{M_{M_{M_{M_{M_{M_{M}}}}}}})$", "$\\psi(\\psi_{M(1,0)}(0))$", "$\\psi(\\psi_{M(1,0)}(1))$", "$\\psi(\\psi_{M(1,0)}(2))$", "$\\psi(\\psi_{M(1,0)}(3))$", "$\\psi(\\psi_{M(1,0)}(4))$", "$\\psi(\\psi_{M(1,0)}(5))$", "$\\psi(\\psi_{M(1,0)}(\\omega))$", "$\\psi(\\psi_{M(1,0)}(\\omega+1))$", "$\\psi(\\psi_{M(1,0)}(\\omega+2))$", "$\\psi(\\psi_{M(1,0)}(\\omega+3))$", "$\\psi(\\psi_{M(1,0)}(\\omega+4))$", "$\\psi(\\psi_{M(1,0)}(\\omega+5))$", "$\\psi(\\psi_{M(1,0)}(\\Omega))$", "$\\psi(\\psi_{M(1,0)}(M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+1))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+\\omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+\\Omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)+M_5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)6))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)\\omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)M_5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^2))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^3))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^4))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^5))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^6))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^\\omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^\\Omega))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^M))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^2}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^3}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^4}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^5}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}))$", "$\\psi(\\psi_{M(1,0)}(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}}))$", "$\\psi(M(1,0))$", "$\\psi(M(1,0)+1)$", "$\\psi(M(1,0)+2)$", "$\\psi(M(1,0)+3)$", "$\\psi(M(1,0)+4)$", "$\\psi(M(1,0)+5)$", "$\\psi(M(1,0)2)$", "$\\psi(M(1,0)3)$", "$\\psi(M(1,0)4)$", "$\\psi(M(1,0)5)$", "$\\psi(M(1,0)^2)$", "$\\psi(M(1,0)^3)$", "$\\psi(M(1,0)^4)$", "$\\psi(M(1,0)^5)$", "$\\psi(M(1,0)^6)$", "$\\psi(M(1,0)^{M(1,0)})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}})$", "$\\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}}}}})$", "$\\psi(M(1,1))$", "$\\psi(M(1,2))$", "$\\psi(M(1,3))$", "$\\psi(M(1,4))$", "$\\psi(M(1,5))$", "$\\psi(M(1,6))$", "$\\psi(M(1,\\omega+1))$", "$\\psi(M(1,\\Omega))$", "$\\psi(M(1,M))$", "$\\psi(M(1,M(1,0)))$", "$\\psi(M(1,M(1,M(1,0))))$", "$\\psi(M(1,M(1,M(1,M(1,0)))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,0))))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,M(1,0)))))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,M(1,M(1,0))))))))$", "$\\psi(M(1,M(1,M(1,M(1,M(1,M(1,M(1,M(1,0)))))))))$", "$\\psi(M(2,0))$", "$\\psi(M(2,1))$", "$\\psi(M(2,2))$", "$\\psi(M(2,3))$", "$\\psi(M(2,4))$", "$\\psi(M(2,5))$", "$\\psi(M(2,\\omega))$", "$\\psi(M(2,\\Omega))$", "$\\psi(M(2,M))$", "$\\psi(M(2,M(1,0)))$", "$\\psi(M(2,M(1,M(1,0))))$", "$\\psi(M(2,M(1,M(1,M(1,0)))))$", "$\\psi(M(2,M(1,M(1,M(1,M(1,0))))))$", "$\\psi(M(2,M(2,0)))$", "$\\psi(M(2,M(2,M(2,0))))$", "$\\psi(M(2,M(2,M(2,M(2,0)))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,0))))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,M(2,0)))))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,M(2,M(2,0))))))))$", "$\\psi(M(2,M(2,M(2,M(2,M(2,M(2,M(2,M(2,0)))))))))$", "$\\psi(M(3,0))$", "$\\psi(M(4,0))$", "$\\psi(M(5,0))$", "$\\psi(M(6,0))$", "$\\psi(M(7,0))$", "$\\psi(M(M,0))$", "$\\psi(M(M(1,0),0))$", "$\\psi(M(M(M(1,0),0),0))$", "$\\psi(M(M(M(M(1,0),0),0),0))$", "$\\psi(M(M(M(M(M(1,0),0),0),0),0))$", "$\\psi(M(M(M(M(M(M(1,0),0),0),0),0),0))$", "$\\psi(M(M(M(M(M(M(M(1,0),0),0),0),0),0),0))$", "$\\psi(M(M(M(M(M(M(M(M(1,0),0),0),0),0),0),0),0))$", "$\\psi(M(1;0)^{M(1;0)})$", "$\\psi(M(1;0)^{M(1;0)}+1)$", "$\\psi(M(1;0)^{M(1;0)}+2)$", "$\\psi(M(1;0)^{M(1;0)}+3)$", "$\\psi(M(1;0)^{M(1;0)}+4)$", "$\\psi(M(1;0)^{M(1;0)}+5)$", "$\\psi(M(1;0)^{M(1;0)}+M)$", "$\\psi(M(1;0)^{M(1;0)}+M_2)$", "$\\psi(M(1;0)^{M(1;0)}+M_3)$", "$\\psi(M(1;0)^{M(1;0)}+M_4)$", "$\\psi(M(1;0)^{M(1;0)}+M_5)$", "$\\psi(M(1;0)^{M(1;0)}+M_6)$", "$\\psi(M(1;0)^{M(1;0)}+M(1,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,1))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,2))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,3))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,4))$", "$\\psi(M(1;0)^{M(1;0)}+M(1,5))$", "$\\psi(M(1;0)^{M(1;0)}+M(2,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(3,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(4,0))$", "$\\psi(M(1;0)^{M(1;0)}+M(5,0))$", "$\\psi(M(1;0)^{M(1;0)}2)$", "$\\psi(M(1;0)^{M(1;0)}3)$", "$\\psi(M(1;0)^{M(1;0)}4)$", "$\\psi(M(1;0)^{M(1;0)}5)$", "$\\psi(M(1;0)^{M(1;0)}\\omega)$", "$\\psi(M(1;0)^{M(1;0)}M)$", "$\\psi(M(1;0)^{M(1;0)}M(1,0))$", "$\\psi(M(1;0)^{M(1;0)}M(2,0))$", "$\\psi(M(1;0)^{M(1;0)}M(3,0))$", "$\\psi(M(1;0)^{M(1;0)}M(4,0))$", "$\\psi(M(1;0)^{M(1;0)}M(5,0))$", "$\\psi(M(1;0)^{M(1;0)+1})$", "$\\psi(M(1;0)^{M(1;0)+2})$", "$\\psi(M(1;0)^{M(1;0)+3})$", "$\\psi(M(1;0)^{M(1;0)+4})$", "$\\psi(M(1;0)^{M(1;0)+5})$", "$\\psi(M(1;0)^{M(1;0)2})$", "$\\psi(M(1;0)^{M(1;0)3})$", "$\\psi(M(1;0)^{M(1;0)4})$", "$\\psi(M(1;0)^{M(1;0)5})$", "$\\psi(M(1;0)^{M(1;0)^{2}})$", "$\\psi(M(1;0)^{M(1;0)^{3}})$", "$\\psi(M(1;0)^{M(1;0)^{4}})$", "$\\psi(M(1;0)^{M(1;0)^{5}})$", "$\\psi(M(1;0)^{M(1;0)^{M}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1,0)}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}}}})$", "$\\psi(M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)^{M(1;0)}}}}}}})$", "$\\psi(M(1;1))$", "$\\psi(M(1;1)2)$", "$\\psi(M(1;1)3)$", "$\\psi(M(1;1)4)$", "$\\psi(M(1;1)5)$", "$\\psi(M(1;1)^2)$", "$\\psi(M(1;1)^3)$", "$\\psi(M(1;1)^4)$", "$\\psi(M(1;1)^5)$", "$\\psi(M(1;1))$", "$\\psi(M(1;2))$", "$\\psi(M(1;3))$", "$\\psi(M(1;4))$", "$\\psi(M(1;5))$", "$\\psi(M(1;\\omega))$", "$\\psi(M(1;M))$", "$\\psi(M(1;M(1;0)))$", "$\\psi(M(1;M(1;M(1;0))))$", "$\\psi(M(1;M(1;M(1;M(1;0)))))$", "$\\psi(M(1;M(1;M(1;M(1;M(1;0))))))$", "$\\psi(M(1;1,0)$", etc. ] Tar(n) = f_{\underbrace{C(C(\cdots(C(C(\Omega_{n}2,0),0),\cdots ),0)}_{n\text{ C's}}}(n) Example of an LNGI/Sequence LaTeX codes: Ultimate List of Large Numbers and Ordinals: ψ(ψ_Ι(0)) to ψ(ε_{Ξ+1}) and beyond up to ψ(C;C;C;....;C) = Church-Kleene Ordinal (e.g. \begin{array}{l}\psi(\psi_I(0)) \\ \psi(\psi_I(0)+1) \\ \psi(\psi_I(0)+\omega) \\ \psi(\psi_I(0)+\Omega) \\ \psi(\psi_I(0)2) \\ \psi(\psi_I(0)\omega) \\ \psi(\psi_I(0)^2) \\ \psi(\psi_I(0)^\omega) \\ \psi(\psi_I(0)^\Omega) \\ \psi(\psi_I(0)^{\psi_I(0)}) \\ \psi(\psi_I(0)^{\psi_I(0)^{\psi_I(0)}}) \\ \psi(\psi_I(1)) \\ \psi(\psi_I(\omega)) \\ \psi(\psi_I(\Omega)) \\ \psi(\psi_I(\psi_I(0))) \\ \psi(I) \\ \psi(I+1) \\ \psi(I+\omega) \\ \psi(I+\Omega) \\ \psi(I+\psi_I(0)) \\ \psi(I2) \\ \psi(I\omega) \\ \psi(I^2) \\ \psi(I^\omega) \\ \psi(I^\Omega) \\ \psi(I^I) \\ \psi(I^{I^I}) \\ \psi(I_2) \\ \psi(I_2+1) \\ \psi(I_2+\omega) \\ \psi(I_2 2) \\ \psi(I_2^2) \\ \psi(I_2^{I_2}) \\ \psi(I_3) \\ \psi(I_{\omega}) \\ \psi(I_{\Omega}) \\ \psi(I_I) \\ \psi(I(1,0)) \\ \psi(I(1,0)+1) \\ \psi(I(1,0)2) \\ \psi(I(1,0)^2) \\ \psi(I(1,0)^{I(1,0)}) \\ \psi(I(1,1)) \\ \psi(I(2,0)) \\ \psi(I(I,0)) \\ \psi(M) \\ \psi(M+1) \\ \psi(M+\omega) \\ \psi(M+\Omega) \\ \psi(M+I) \\ \psi(M2) \\ \psi(M\omega) \\ \psi(M^2) \\ \psi(M^\omega) \\ \psi(M^\Omega) \\ \psi(M^I) \\ \psi(M^M) \\ \psi(M^{M^M}) \\ \psi(M_2) \\ \psi(M_2+1) \\ \psi(M_2+\omega) \\ \psi(M_2 2) \\ \psi(M_2^2) \\ \psi(M_2^{M_2}) \\ \psi(M_3) \\ \psi(M_{\omega}) \\ \psi(M_{\Omega}) \\ \psi(M_M) \\ \psi(M(1,0)) \\ \psi(M(1,0)+1) \\ \psi(M(1,0)2) \\ \psi(M(1,0)^2) \\ \psi(M(1,0)^{M(1,0)}) \\ \psi(M(1,1)) \\ \psi(M(2,0)) \\ \psi(M(M,0)) \\ \psi(M(M(1,0),0)) \\ \psi(M) \end{array}) Tier 2 The document starts at millillion, to give the audience an overview. Tier 1 Index Tier 2 Index Name Abbreviation 1,000 1 millilllion MI 1,001 milli-unilllion MI-U 1,002 milli-duilllion MI-D 1,003 milli-trilllion MI-T 1,004 milli-quadrilllion MI-Qa 1,010 milli-decilllion MI-De 1,100 milli-centilllion MI-Ce 2,000 dumillilllion DMI 2,001 dumilli-unilllion DMI-U 3,000 trimillilllion TMI 4,000 quadrimillilllion QaMI 10,000 decimillilllion DeMI 11,000 undecimillilllion UDeMI 20,000 vigintimillilllion VgMI 100,000 centimillilllion CeMI Index Name (Myrillions) Abbreviation 10,000 myrillion MY 11,000 myrimillillion MY-MI 20,000 dumyrillion DMY 100,000 decimyrillion DeMY 1,000,000 2 micrilllion MC 1,000,001 micri-unilllion MC-U 1,001,000 micri-millilllion MC-MI 1,001,001 micri-milli-unilllion MC-MI-U 2,000,000 dumicrilllion DMC 10,000,000 decimicrillion DeMC 1,000,000,000 3 nanilllion NA 1,000,000,001 nani-unillion NA-U 2,000,000,000 dunanillion DNA 1,000,000,000,000 4 picillion PI 1,000,000,000,000,000 5 femtillion FM My extension starts at an attillion. So, get ready! (also includes the old version, which is always capitalized) Tier 2 Index Name Abbreviation Old version 5 femtilllion Fem FM 6 attilllion At AT 7 zeptilllion Zep ZP 8 yoctilllion Yo YT 9 rontillion Rt RT 10 quectillion Qu QE 11 mecilllion Me(c) ME 12 duecilllion Du(c) DE 13 trecilllion Tre(c) TE 14 tetrecilllion Te(c) TeE 15 pentecilllion Pe(c) PE 16 hexecilllion He(c) HE 17 heptecilllion Hp(c) HeE 18 octecilllion Ot(c) OC 19 ennecilllion En(c) EC 20 icosilllion Is(o) IS 21 meicosilllion MeIs MS 22 dueicosilllion DuIs DS 23 trioicosilllion TreIs TS 24 tetreicosilllion TeIs TeS 25 penteicosilllion PeIs PS 26 hexeicosilllion HeIs HS 27 hepteicosilllion HpIs HeS 28 octeicosilllion OtIs OS 29 enneicosilllion EnIs ES 30 triacontilllion TrC / TCn TN 31 metriacontilllion MeTrC MTN 32 duetriacontilllion DuTrC DTN 1 Using BIPM’s 2022 proposal. Large quantitative prefixes are ronna and quetta. (formerly quecca) You might see significant differences as I go further from traditional -illions. The abbreviations of unit roots have been changed drastically to avoid ambiguous abbreviations easier. The E letter from 10-17th tier 2 roots has been changed into C. The icosi- root now always stands for IC, regardless of the following tier 2 units. Same goes to the following “tens” roots: triaconta-, tetraconta-, etc. Let’s speed up the work. Tier 2 Index Name Abbreviation Old version 10 quectilllion Qc(t) VE 20 icosilllion Is(o) IS 30 triacontilllion TrC TN 40 tetracontilllion TeC TeC 50 pentacontilllion PeC / PCo PC 60 hexacontilllion HeC / HCt HC 70 heptacontilllion HpC HeC 80 octacontilllion OtC / OCn OC 90 enneacontilllion EnC / ECo EC 100 hectilllion Hec HT 101 mehectilllion MeHec MHT 102 duehectilllion DuHec DHT (limit in NG+3 standalone) 110 quectehectilllion QtHc QEHT 111 mecehectilllion MecHc MEHT 112 duecehectilllion DucHc MEHT 120 icosehectilllion IsHc ISHT 200 dohectilllion DHc DT 210 quectedohectilllion QtDHc QEDT 300 triahectilllion TrH / THt TT 400 tetrahectilllion TeH TeT 500 pentahectilllion PeH / PHc PT 600 hexahectilllion HeH / HHe HT 700 heptahectilllion HpH / HpH HeT 800 octahectilllion OtH / OHt OT 900 enneahectilllion EnH / EHc ET 990 enneconteenneahectilllion EnCEnH ECET 999 enneenneconteenneahectilllion EnEnCEnH EECET That’s the Standard limit in NG+3R formatting. However, it’s not over yet. Now, let’s go to bigger, hyperdimensional names. Tier 1 (Maximus) Before I move on to Tier 3, let’s abbreviate some maximus- numbers. Exponent Maximus Index Name Abbreviation 1,000 0 maximusthousand MXS-K 1,000,000 1 maximusmillion MXS-M 1,000,000,000 2 maximusbillion MXS-B 1,000,000,000,000 3 maximustrillion MXS-T By the way, 10^x = MXS-x. Tier 3 Do you know that in my Standard conversion, it stops at nanepetillion? There are no tier 3 changes, except terillions. Tier 2 Index Tier 3 Name Abbreviation 1,000 1 killillion Kl 1,000+ 1+ killo-milliillion Kl-Mi 1,000* dukillillion DKl 1,001 killa-millillion Kla’Mi 1,002 killa-micrillion Kla’Mc 1,010 killa-quectillion Kla’Qu 1,100 killa-hectillion Kla’Hc 2,000 1* micrekillillion duekillillion McKl DuKl 3,000 nanekillillion triokillillion NaKl TrKl 10,000 quectekillillion QuKl 100,000 hectekillillion HecKl 1,000,000 2 megillion Mg 1,000,001 2+ mega-millillion Mga’Mi 1,001,000 mega-killillion Mga’Ki 2,000,000 2* micremegillion duemegillion McMg DuMg 1,000,000,000 3 gigillion Gi(g) 1,000,000,000,000 4 terillion Tr / Ter 1,000,000,000,000,000 5 petillion Pt 3,000,000,000,000,000 5* nanepetillion triopetillion TrPt For now on, I have to make up new abbreviations to go up to the limit. Tier 3 Name Abbreviation 6 exillion Ex 7 zettillion Zt 8 yottillion Yt 9 ronnillion Rn 10 quettillion Qt 11 hendillion Hn / He 12 dokillion Dok 13 tradakillion TrD 14 tedakillion TeD 15 pedakillion PeD 16 exdakillion ExD 17 zedakillion ZeD 18 yodakillion YoD 19 nedakillion NeD 20 ikillion Ik / IK 21 ikenillion IkeN 22 icodillion IcoD 23 ictrillion IctR 24 icterillion IctE 25 ikpetillion IkpT 26 ikexillion IkeX 27 iczetillion IczE 28 ikyotillion IkyO 29 icronillion IcrN 30 trakillion Trk / TrK 31 trakenillion TkeN / TrkeN 32 tracodillion TcoD / TrkoD 33 tractrillion TctR / TrktR 40 tekillion Tek / TeK 41 tekenillion TekeN 50 pekillion Pek / PeK 51 pekenillion PkeN / PekeN 60 exakillion Exk / ExK 61 exakenillion EkeN / ExkeN 70 zakillion Zak / ZaK / Za 71 zakenillion ZkeN / ZakeN 80 yokillion Yok / YoK 81 yokenillion YkeN / YokeN 90 nekillion Nek / NeK 91 nekenillion NkeN / NekeN 99 necronillion NcrN / NecrN 100 hotillion HoT 101 hotenillion HoEN 102 hotodillion HoOD 103 hotrillion HoTR 104 hoterillion HoTE 105 hopetillion HoPT 106 hotectillion HoEC 107 hozetillion HoZT 108 hoyotillion HoYT 109 horonnillion HoRN 110 hoquettillion HoQT 111 hotendillion HoEn 112 hodokillion HoDoK 113 hotradakillion HoTrD 114 hotedakillion HoTeD 115 hopedakillion HoPeD 116 hoexdakillion HoExD 117 hozedakillion HoZeD 118 hoyodakillion HoYoD 119 honedakillion HoNeD 120 hotikillion HoIK 121 hotikenillion HoIkeN 130 hotrakillion HoTrK 160 hotexakillion HoTeK 200 dotillion DoT 300 totillion ToT 400 trotillion TrT 500 potillion PoT 600 exotillion ExT 700 zotillion ZoT 800 yootillion YoT 900 notillion NoT 990 nonekillion NoNK 999 nonecronillion NoNcrN Time to blast off to the astronomical step! Tier 4 Now, we are approaching outer space. We will now see the hypercelestial objects... The machine seems to collect the celestial dust and create more powerful abbreviations. After months of break on this document, I am finally introducing Tier 4! You know the drill, KaL = kalillion, MeJ = mejillion, GiJ = Gijillion, etc. Remove the last letter but except coming before eN, oD, and tR. Tier 3 Name Abbreviation 1,000 kalillion KaL 1,001 kalenillion KaLeN 1,002 kalodillion KaLoD 1,003 kaltrillion KaLtR 1,004 kalterillion KaTer 1,005 kalpetillion KaPt 1,006 kalexillion KaEx 1,007 kalzettillion KaZt 1,008 kalyottillion KaYt 1,009 kalronnillion KaRn 1,010 kalquettillion KaQu 1,011 kalhendillion KaHn 1,020 kalikillion KaIk 1,030 kaltrakillion KaTrk 1,100 kalhotillion KaHoT 1,999 kalnonecronillion KaNoNcrN 2,000 dalillion DaL 2,001 dalenillion DaLeN 2,010 daldakillion DaDk 2,100 dalhotillion DaHoT 3,000 talillion TaL 4,000 tralillion TraL 5,000 palillion PaL 6,000 exalillion ExaL 7,000 zalillion ZaL 8,000 yalillion YaL 9,000 nalillion NaL 10,000 dakalillion DKaL 11,000 hendalillion HNaL 20,000 ikalillion IKaL 100,000 hotalillion HoTaL 1,000,000 mejillion MeJ 1,000,001 mejenillion MeJeN 1,001,000 mejkalillion MeKaL 1,010,000 mejdakalillion MeDKaL 2,000,000 dejillion DeJ 1,000,000,000 gijillion GiJ 2,000,000,000 dijillion DiJ Tier 4 Name Abbreviation* 4 astillion AsT 5 lunillion LuN 6 fermillion FrM 7 jovillion JoV 8 solillion SoL 9 betillion BeT** 10 glocillion GlO 11 gaxillion GaX 12 supillion SuP 13 versillion VrS 14 multillion MlT After gijillion, add ` instead of removing the last letter in a process. ** When tier 3 roots come before betillion, replace e with ee to follow Douglas’ fixes. [Extended Tier 4 coming soon] although barrillion = BrRl, barrekalillion = BrKaL [Tier 5 coming soon] although hepillion = HepA, geopillion = GepA [Tier 6 coming soon] although hapaxillion = HPx, tetrakisillion = TeKs [Tier 7 coming soon] although redillion = Rd, icoise-resepi-decakisillion = IseRp-DcKs [[ INTRODUCTION ]] From early history, humans thought of an extensible naming system for grouping large numbers. That is, in English, we group with millions, billions, and trillions. Around the 20th century, mathematicians proposed new illions that haven't been seen from the real world before. In the early 2000s, Bowers decided to start his extensive list of illions, including numbers that reach beyond the realistic range. Sbiis, on the other hand, released the extensible system for Bowers’ -illions, with all naming gaps filled out. Sbiis’ version Out of curiosity, Douglas pointed out there was a small mistake that the petillion is ambiguous from his list. He fixed the problem by putting another e for tier 3 multipliers to -et root. He also published the LNGI page for Sbiis’ illions which is shown below. Numbers Getting Bigger - illions version Nearly a year after the reformation began, DeepLineMadom fixed the ambiguity problem for yocillion too! The earlier confusing -illions were fixed by adding “i” in-between several root connections. With problems fixed, we could name numbers up to the limit of Bowers’ illions, which is the supremum of all tier 0 - 4 roots being used. In this page, I am going to take Sbiis’ and Douglas’ spirits and continue the journey of never-ending illions list. Get ready, let’s reach out of the ethereal illions! [EXTENSIONS] Rise out of Bower’s illions, there are joints to different possibilities! One is spreading out to a tetrational range which metaness can’t keep up! In early 2020s, Nirvana (now CompactStar) decided to extend, up to the ridiculous limit of the Tier 6 system! Regardless of only milestones, some existing names are gathered for amalgamates… CompactStar’s -illions Not only that, there’s many illions going after multillion, like metillion, pyrillion, metalillion, and so infinitum. Tier 5: hepillion / febrillion / keltillion / wektillion Tier 6: hapaxillion Tier 7: redillion Tier 8: erkillion Tier 9: izzillion Tier 10: meakoowillion However, I will only fill out his system and some more, even going beyond Tier 6! I will also fix his Tier 5 system too because of the predefined connections too. (geopahepillion is already precoined, by [Tier 2] connections between geopillion and hepillion) (Tier 2 additive: geop + (a) + hep -> geopahepillion) With connections ready, let’s aim at the naming gaps. [ILLION FUNCTION] Unfortunately, the Extensible Illion System is ill-defined. I have to define a new function by myself: Ha = Ha#1 Ha#0 = 1,000*a b > 0: Ha#b = H(1,000^a)#(b-1) Examples: H34 = H34#1 = H(1,000^34)#0 = 1,0001,000^34 = 10^105 = 1 quattourtrigintillion H2,013#2 = H(1,000^2,013)#1 = 1,000^(1,000^2,013+1) = 10^(310^6,039+3) = 1 micrekillatrecillion Illion Function can also be expressed in Hyper-E: Ha#b = 1,000E(1,000)a#b And can be estimated approximately: 1,000^^(b+1) > Ha#b > 1,000^^b (for a < 333) Yes, this can be tweaked into a variant of Sbiis’ Extensible-E Notation. All it needs is a simple rule. I denote E as an ExE base. Hn -> 1,000*E(1,000)n [[ LIST OF ILLIONS ]] [Tier 4] - HYPERCELESTIAL :[ 10^10^10^10^45 - 10^10^10^10^3000 ]: :[ IMPROVED BY DEEPLINEMADOM, FIXED BY AAREX ]: Hang on… How I start at metillion? Guess what, the naming gaps were already filled. I was going meta for nonstandard illions whatsoever. But, one ubiquitous issue. If I take met and put the first letter off, it would result the same as I do to bet. They are splitted into 2 identical "et." (taking the first letter out: bet -> et, met -> et) But did I stump on that? NO. I want to do a similar resolution as what Douglas resolved. He decided to turn et and eet when the first letter (b) is removed. This is consistent as you are adding Tier 3 multiplicatives to Tier 4. (1st: Taking the first letter out: bet -> et, 2nd: Douglas’ resolution, adding another e: et -> eet, 3rd: Finally, add Tier 3 roots as multiplicative: hot + eet -> hoteet) This doesn’t work for other tier multiplicatives and all additives: (Tier 1 multiplicative: vigint + (i) + bet -> vigintibet) (Tier 2 multiplicative: duec + (e) + bet -> duecebet) (Tier 1 additive: bet + (o) + zept -> betozept) (Tier 2 additive: bet + (a) + xenn -> betaxenn) (Tier 3 additive: bet + mej -> betmej) I eradicate met into ett for Tier 3 multiplicatives to met. In that case, take out the first letter and add another t after e. (d + met - (m) + (t) -> dettillion) (tr + met - (m) + (t) -> trettillion) (t + met - (m) + (t) -> tettillion) (sequence: metillion, dettillion, trettillion, tettillion, …) Miserly, one collides with the existing zettillion… (z + met - (m) + (t) -> zettillion: collision detected) So the resolved variant for zettillion has to be degenerated to zeettillion. Just add another e, like Douglas! (z + (e) + met - (m) + (t) -> zeettillion) Now I can go hypercelestial and reach far out of natural cosmology! Let's blast at speeds of abstraction! Tier 3 Index Number Name Tier 3 Index Number Name 1027 H(110^27)#3 10^(310^(310^(310^27))+3) betillion 1045 H(110^45)#3 10^(310^(310^(310^45))+3) metillion 21027 H(210^27)#3 10^(310^(310^(610^27))+3) deetillion 21045 H(210^45)#3 10^(310^(310^(610^45))+3) dettillion 31027 H(310^27)#3 10^(310^(310^(910^27))+3) treetillion 31045 H(310^45)#3 10^(310^(310^(910^45))+3) trettillion 41027 H(410^27)#3 10^(310^(310^(1210^27))+3) teetillion 41045 H(410^45)#3 10^(310^(310^(1210^45))+3) tettillion 51027 H(510^27)#3 10^(310^(310^(1510^27))+3) peetillion 51045 H(510^45)#3 10^(310^(310^(1510^45))+3) pettillion 61027 H(610^27)#3 10^(310^(310^(1810^27))+3) exeetillion 61045 H(610^45)#3 10^(310^(310^(1810^45))+3) exettillion 71027 H(710^27)#3 10^(310^(310^(2110^27))+3) zeetillion 71045 H(710^45)#3 10^(310^(310^(2110^45))+3) zeettillion (resolved) 81027 H(810^27)#3 10^(310^(310^(2410^27))+3) yeetillion 81045 H(810^45)#3 10^(310^(310^(2410^45))+3) yettillion 91027 H(910^27)#3 10^(310^(310^(2710^27))+3) neetillion 91045 H(910^45)#3 10^(310^(310^(2710^45))+3) nettillion 101027 H(1010^27)#3 10^(310^(310^(3010^27))+3) dakeetillion 101045 H(1010^45)#3 10^(310^(310^(3010^45))+3) dakettillion 111027 H(1110^27)#3 10^(310^(310^(3310^27))+3) hendeetillion 111045 H(1110^45)#3 10^(310^(310^(3310^45))+3) hendettillion Welcome to the next level… The list of illions go wild out! o_0` Now eradicate the first letters and mutate it on higher roots! This goes legionically worse; Root 0s 10s Multiplied Root 0s 10s +0 gloc +0 -oc +1 kal gax +1 -al -ax +2 mej sup +2 -ej -up +3 gij vers +3 -ij -ers +4 ast mult +4 -ast -ult +5 lun met +5 -un -ett +6 ferm xev +6 -erm -ev +7 sol hyp +7 -ol -(oh)yp +8 jov omni(v) +8 -ov -omni(v) +9 bet out +9 -eet -ut [1] With omnillion / omnivillion consensus, having a (v) is optional. [2] (oh) is optional too. Onto the local scale of barrels and glocs (globular cluster); Repeat the process for placing a tier 3 multiplicative by removing the first letter. Besides with Tier 3 multiplicatives to Tier 4… Let's fold the Tier 4 roots by generalizations. It has been recently tweaked to use most of the tier 4 factors instead. Root Ones Tens 0 (use the base roots instead) 1 -al 2 -ej barr- 3 -ij gic- 4 -ast* asc-* 5 -un luc- 6 -erm ferc- 7 -ov joc- 8 -esol solc-* 9 -et bec- [1]* The 24th tier-4 root is actually barreast. “Barrast” is ambiguous in Fonster’s system. [2] “-esol” remains unchanged for pronunciation. [3] Like astillion, do not remove “a” from asc-. [4] soc- is renamed to solc- due to collision to joc- with tier 3 multiplicatives. This allows us to name illions up to exactly cetillion, at 100th Tier 4! Using Bower’s polytope naming with generalized Compact Star’s -illions, the highest dimension before polyhuton is named: poly nonacxenecet-nonacxenecesol-nonacxenecov- nonacxenecerm-nonacxenecun-nonacxenecast- nonacxenecij-nonacxenecej-nonacxenecal- nonacxenec- nonacxenolcet-nonacxenolcesol-nonacxenolcov-- ... ... ... nonacxenarr-nonacxenut-nonacxenomn- nonacxenohyp-nonacxenev-nonacxenett- nonacxenult-nonacxeners-nonacxenup- nonacxenax- nonacxenoc-nonacxeneet-nonacxenol- nonacxenov-nonacxenerm-nonacxenaun- nonacxenast-nonacxenij-nonacxenej- nonacxenal-nonacxenon (10300-1-dimensional polytope) Let's take a major step up from fictional cosmology, and observe the farthests. "Cet Em' Down, Jut no Buts!" ~ Proto-Alien from Type 9.7 Archverse "Lut the Hundreds up, delete the (T)s, and Gut the Ones and Tens on the right! That's we build the Huts of Hypercelestial Scales!" “And let the Celestial Entity fix the “t” conflict after hundreds!” This is extraordinary… If you didn't understand, check the root table below: Root Ones (+10) [1] (+tens) [1] Tens Hundreds [2] 0 gloc (use the base roots instead) 1 kal gax al cet 2 mej sup ej barr mejet2 3 gij vers ij gic git 4 ast mult east2 asc aset 5 lun met un luc lut 6 ferm xev erm ferc fert 7 jov hyp ov joc jovt 8 sol omn(iv) sol/esol1 solc1 solt2 9 bet out et bec bect2 Tier 3 Connection [3] Ones Ones (+10) Tens Hundreds 0 oc (use the base roots instead) 1 al ax ocet*1 2 ej up arr ejet 3 ij ers ic it 4 ast ult asc aset 5 un eet uc unt 6 erm ev erc ert 7 ov ohyp oc ovt*2 8 ol omni olc olt 9 et ut ec ect*2 [1] Use (+10) for 10th - 19th teens roots. Else, use (+tens). [2] Remove the final “t” if it follows with “ones” or “tens” roots. [3] Only if it precedes with Tier 3 root. [*1] Renamed due to pronunciation [*2] Renamed due to conflicts. Examples (201) mejekalillion (410) aseglocillion (621) ferbarralillion (300) gitillion (700) jovtillion Now we organize the huts of hundreds, here’s the list of all tier-4 -illions! Number Tier 3 Index Tier 4 Index Name H1#4 = 10^(310^(310^3,000))+3) 1,000 1 kalillion H(10^3,000+1)#2 = 10^(310^(310^3,000+3))+3) 1,000+ 1+ kala-killillion H(210^3,000)#2 = 10^(310^(6*10^3,000))+3) 1,000+ duekalillion / micrekalillion H1,001#3 = 10^(310^(310^3,003))+3) 1,001 kalenillion H1,002#3 = 10^(310^(310^3,006))+3) 1,002 kalodillion H1,003#3 1,003 kaltrillion H1,004#3 1,004 kalterillion 1,010 kaldakillion 1,011 kaltendillion 1,020 kalikillion 1,100 kalhotillion H2,000#3 2,000 1* dalillion 2,001 dalenillion 3,000 tralillion H10,000#3 10,000 dakalillion 11,000 hendalillion 20,000 ikalillion H100,000#3 100,000 hotalillion H2#4 = H1,000,000#3 = 10^(310^(310^3,000,000))+3) 1,000,000 2 mejillion 1,000,001 2+ mejenillion 1,001,000 mejkalillion 1,001,001 mejkalenillion 1,002,000 mejdalillion 2,000,000 2* dejillion 10,000,000 dakejillion 100,000,000 hotejillion H3#4 = 10^(310^(310^(3*10^9)))+3) 1,000,000,000 3 gijillion 1,001,000,000 3+ gijmejillion H4#4 = 10^(310^(310^(3*10^12)))+3) 1,000,000,000,000 4 astillion 1,001,000,000,000 4+ astgijillion H5#4 = 10^(310^(310^(3*10^15)))+3) 1,000,000,000,000,000 5 lunillion 1016 5* dakunillion H6#4 1018 6 fermillion 1019 6* dakermillion H7#4 1021 7 jovillion 1022 7* dakovillion H8#4 1022 8 solillion 1024 8* dakolillion H9#4 1025 9 betillion 1027 9* dakeetillion H10#4 1028 10 glocillion 1030 10* dakocillion H11#4 1031 11 gaxillion 1033 11 dakaxillion H12#4 1034 12 supillion 1036 12 dakupillion H13#4 1037 13 versillion 1039 13 dakersillion H14#4 1042 14 multillion 1043 14 dakultillion 9991042 14 nonecxenultillion H15#4 1045 15 metillion 1046 15* dakettillion H16#4 1048 16 xevillion 1049 16* dakevillion H17#4 1051 17 hypillion 1052 17* dakypillion H18#4 1054 18 omnillion 1055 18* dakomnillion H19#4 1057 19 outillion 1058 19* dakoutillion H20#4 1060 20 barrillion 1061 20* dakarrillion H21#4 1063 21 barralillion H22#4 1066 22 barrejillion H23#4 1069 23 barrijillion H24#4 1072 24 barrastillion H25#4 1075 25 barrunillion H26#4 1078 26 barrermillion H27#4 1081 27 barrovillion H28#4 1084 28 barresolillion H29#4 1087 29 barretillion H30#4 1090 30 gicillion 1091 30* dakicillion H31#4 1093 31 gicalillion H32#4 1096 32 gicejillion H33#4 10120 40 ascillion 10121 40* dakascillion H30#4 10150 50 lucillion 10151 50* dakucillion H60#4 10180 60 fercillion 10181 60* dakercillion H70#4 10210 70 jocillion H80#4 10240 80 solcillion H90#4 10270 90 becillion H99#4 10297 99 becetillion H100#4 10300 100 cetillion 10301 100* dakocetillion H101#4 10303 101 cekalillion H102#4 10306 102 cemejillion H110#4 10330 110 cegaxillion H120#4 10360 120 cebarrillion H121#4 10363 121 cebarralillion H130#4 10390 130 cegicillion H140#4 10420 140 ceascillion H200#4 10600 200 mejetillion 10601 200* dakejetillion H300#4 10900 300 gitillion H400#4 101,200 400 asetillion H500#4 101,500 500 lutillion H600#4 101,800 600 fertillion H700#4 102,100 700 jovtillion H800#4 102,400 800 soltillion H900#4 102,700 900 bectillion H990#4 102,970 990 becbecillion H999#4 102,997 999 becbecetillion Before we get extraordinary, let's peek into other proposals for extended Bowers' illions. [Other Systems] There's so many possibilities from obscure websites. They are even organized in a sheet. It is always fun, so why not explain in this page? There is another name for 15th tier-4 -illion which is familiar to most googologists: pyrillion. I think this is a combination of pyro and -illion, but let’s get over it. Intermediate googologists just remove “p” for tier 3 multiplicatives. This is what I did: “The second letter is y, so I use the hypillion method. Just add an o- at the left of pyr- root!” This coins dopyrillion, which is 2*1045th tier-3 -illion, and several others. But, is that it? Wrong. There are so many googologisms left to go. Here are the possibilities for tier 4. Tier 4 Index Possibility 1 Possibility 2 Possibility 3 (by MultiSoul) Possibility 4 15 pyrillion (met) metalillion 16 guntillion xenillion xenoxillion 17 kentrillion hypeillion hyperillion 18 onlillion (omni) omnivillion 19 paptrillion supeillion (not confused with supillion) outsillion 20 housillion dupeillion (not confused with dupillion) barrelillion idixillion 21 horsillion / housalillion barrelalillion 22 barrelejillion 23 barrelijillion 24 barrelestillion (barrelastillion is predefined as tier-3 -illion) 25 barrelunillion 26 barrelermillion 27 barrelovillion 28 barrelolillion 29 barreletillion 30 trongillion garrelillion gradixillion 40 batillion astarrelillion asixillion 50 handrillion lunarrelillion lenixillion 60 mastillion fermarrelillion ferixillion 70 gotillion jovarrelillion javixillion 80 kortillion solarrelillion solixillion 90 nenkillion (from other site) Possibility 5 betarrelillion betixillion 100 janillion horillion huutillion 200 mejanillion morillion muutillion 300 gijanillion guutillion 400 astanillion astuutillion 500 fermanillion luutillion 600 lunanillion fuutillion 700 jovanillion juutillion 800 solanillion suutillion 900 betanillion buutillion Sigh, here I go. Possibility 1 is easy to generalize, because you can use these words, previous tier 4 multiplicatives (*even to marillions), to make intermediate values. Root Ones Ones (+10) Ones (+20) Tens Hundreds Root (tier 3 multiplicatives / tier 4 additives) Ones Ones (+10) Tens Hundreds 0 gloc (use the base roots instead) 0 -oc (use the base roots instead) 1 kal gax hors (optional) jan 1 -al -ax -an 2 mej sup hous mejan 2 -ej -up -ous -ejan 3 gij vers trong gijan 3 -ij -ers -ong -ijan 4 ast mult bat astan 4 -(a/e)st -ult -at -astan 5 lun pyr handr lunan 5 -un -(o/e)pyr -andr -unan 6 ferm gunt mast ferman 6 -erm -unt -(o/e)mast (to avoid with -ast) -erman 7 jov kentr got jovan (to avoid with jar) 7 -ov -entr -(o/e)got (to avoid with -ot) -ovan 8 sol onl kort solan 8 -ol -eonl -ort -olan 9 bet paptr nenk betan 9 -eet / -et -aptr -enk -etan The limit for Possibility 1 is betanenketillion. I made up for Possibility 1a, for using tier 4 multiplicatives to marillions for “hundreds” roots. Root Hundreds Hundreds (tier 3 multiplicatives) 0 1 jan -an 2 mejan -ejan 3 gan -ogan 4 asan -asan 5 lan -olan 6 fan -ofan 7 jovan (to avoid with jar) -ovan 8 san -olan 9 ban -etan Same difficulty goes to Possibility 3. Root Ones Ones (+10) Tens Hundreds Root (tier 3 multiplicatives / tier 4 additives with 10s) Ones Ones (+10) Tens Hundreds 0 gloc (use the base roots instead) 0 -oc (use the base roots instead) (add the o at the left) 1 kal gax huu(t) 1 -al -ax 2 mej sup barrel muu(t) 2 -ej -up -arrel 3 gij vers garrel guu(t) 3 -ij -ers -ogarrel 4 ast mult astarrel asuu(t) 4 -(a/e)st -ult -astarrel 5 lun metal lunarrel luu(t) 5 -un -etal -unarrel 6 ferm xenov fermarrel fuu(t) 6 -erm -enov -ermarrel 7 jov hyper jovarrel juu(t) 7 -ov -ohyper -ovarrel 8 sol omniv solarrel suu(t) 8 -ol -omniv -olarrel 9 bet outs betarrel buu(t) 9 -eet / -et -uts -etarrel The limit for Possibility 3 is buubetarreletillion. Possibilities 2 are just milestone proposals, so it is necessary to fill the naming gap up to hutillion. [WIP] (34) truperestillion Possibility 4 is just alternate names for tier 4 “ten” roots, so the 21st Tier-4 -illion with Possibility 4 is idixekalillion. Possibility 5 is just names for 100th and 200th, but it is incomplete. This can be generalized by using the first syllable in Compact Star's hundred roots and removing "r" for additive roots. (826) sobarrermillion Here are the “999” limits for all 5 Possibilities. Underlined means it is changed to match with a respective Possibility. Possibility 1: betanenketillion Possibility 2: bunonuperetillion Possibility 3: buubetarreletillion Possibility 4: bubetixetillion Possibility 5: borbetarreletillion [MADOM POSSIBILITY: 5/27/22] I shout out to DeepLineMedom for making a more reasonable Tier 4 system, with less hyper-cosmological and more Bowersian. The tradition there doesn’t stop abruptly, with an existing language being used like Tier 2 and Tier 3. https://sites.google.com/view/deeplinemadoms-googology/illion-system/first-attempt https://sites.google.com/view/deeplinemadoms-googology/illion-system/remake-2 Note that my new take will now use Madom’s attempts, which I use attempt 1 up to hundreds and remake 2 for hundreds. Also, there are some tweaks to reconstruct my new -illion system, also using Saibian's nomenclature of the form E100{#,#+1,1,2}n. Root Ones Ones (+10) Tens Hundreds 0 gloc (use the base roots instead) 1 kal gax kent 2 mej sup twe deuxent 3 gij vers friel troisent 4 ast mult fioril quaternt 5 lun met finn cinquent 6 ferm xev sexil sixent 7 jov hyp sjour septent 8 sol omni(v) attal huitent 9 bet box neiul neufent Root (multiplied by tier 3) Ones Ones (+10) Hundreds (+tens) 0 -oc 1 -al -ax -ent 2 -ej -up -euxent 3 -ij -ers -oisent 4 -ast -ult -atrent 5 -un -ett -inquent 6 -erm -ev -ixent 7 -ov -ohyp -eptent 8 -ol -omni -uitent 9 -eet -ox -eufent Add “o” to connect with tier 3 multiplicative and tier 4 tens / hundreds root. Add “i” inbetween to construct one-ten combinations. Alternatively, you can also use base-20 with all “one” roots for every hundred. (33) versitweillion is valid, and so gijifrielillion. [CHANGES FROM VERSION 1] Formerly, Tier 3 multiplicatives to -out do not overlay on tier-4 hundreds. It has been changed. Tier 3 Index Name Tier 3 Index Name 1057 outillion 10300 hutillion 21057 dutillion 10600 mutillion 31057 trutillion 10900 gutillion 41057 tutillion 101,200 astutillion 51057 putillion 101,500 lutillion 61057 exutillion 101,800 futillion 71057 zutillion 102,100 sutillion 81057 yutillion 102,400 jutillion 91057 nutillion 102,700 butillion Alternatively, you can keep “o” as it is except hutillion. It’s optional to mind the letters out. The 40th Tier-4 illion was called “aarrillion” before DeepLineMadom made it more contingent. Besides with the connections between units and tens, I split from GeneralStar’s ones. Root Ones Tens 0 (use the base roots instead) 1 -ekal 2 -ej barr- 3 -ij gar[r]- 4 -[e]ast* aarr- 5 -un lunarr- 6 -erm fermar[r]- 7 -ov solarr- 8 -esol jovar[r]- 9 -et betarr- [CHANGES FROM VERSION 2] Anything in Version 2 and prior ones were made from CompactStar’s first version. Root Ones Ones (+10) Tens Hundreds Root (after hundreds) Post-100s Fix 0 gloc (use the base roots instead) 1 tal 1 kal gax hu(t) 2 tej 2 mej sup barr mu(t) 3 tij 3 gij vers gar[r] gu(t) 4 tast 4 ast mult aarr as[t]u(t) 17 typ 5 lun met lunarr lu(t) 18 tomn 6 ferm xev fermar[r] fu(t) 19 tout 7 jov hyp jovar[r] ju(t) 40 tastarr 8 sol omni(v) solarr su(t) 9 bet out betarr bu(t) Root (multiplied by tier 3) Ones Ones (+10) Tens Hundreds 0 -oc (use the base roots instead) (add the o at the left) 1 -al -ax 2 -ej -up -obarr 3 -ij -ers -ogar[r] 4 -ast -ult -oaarr 5 -un -ett -unarr 6 -erm -ev -ermar[r] 7 -ov -ohyp -ovar[r] 8 -ol / -esol -omni -olarr 9 -eet / -et -ut -etarr Use “option 2” if it is followed by tens. Remove (t) if it comes before “tens” / “ones” roots. [r/t] are optional to remove. If there’s a root found in “Post-100 Fix,” replace it if it is followed by a “hundreds” root. [Tier 5] - HOAXORDINARY :[ 10^10^10^10^3000 - 10^10^10^10^10^3000 ]: Tier 5 is the first tier not shown in Bowers’ original list. Let’s keep pushing towards the summit! unvillion = z(1, 5) [1,000th tier-4] I’ll use Sbiis’ system whereas I use “i” connection to add Tier 5 root with Tier 3. (tier 4 roots alone are okay with Fonster’s) Check the original source by Madom. (e.x. unvimeg = 103,000+2nd tier-3) “Thought I’m done with Tier 5 reformations?” Gaps are filled by CompactStar except it’s not doing well. I generalize that “u” is used to add Tier 5 with a Tier 4 root. By word magic, we can remove some letters for shortening reasons. Here’s the multiplicatives. Root Ones (+10) (+tens) Tens 0 glo(c) (unchanged) (unchanged except barr) 1 k ga 2 m su bar(r)[1] 3 g ver gic 4 a mul asc 5 l me luc 6 f xe ferc 7 j hy joc 8 s omn solc 9 b ou bec [1] (r) is required to connect with “+tens” roots. For Tier 3 multiplicatives, add o- inbetween. “Hundred” roots remain unchanged. Anyways, reveal the fifth tier root lookup! Root Ones Ones (+10) Tens Hundreds 0 ek (use “ones” for pre-13) 1 unv ev ek evaf 2 duv twyv lan duvaf 3 et etk etaf 4 cuv cuk cuvaf 5 quiv quik quivaf 6 swiv swik swivaf 7 pryv pryk pyrvaf 8 oiv oik oivaf 9 en enk envaf Tier 4 multiplicatives: [1] For duv, remove “d.” [2] For tier-5 roots that doesn’t start with “u,” add “u” inbetween. (e.x. muekillion = 21030 tier-4 illion = z(21030, 4)) Seeing from the luminosity of magic, the system favors the mistakes. "Knowledge from Sbiis’ hoax prefixes inspires the fix for these roots. From there, the letters were eradicated with some conflict fixes. A magical change, and the roots blew up like an explosion!" /// /// [ NUMBER LIST COMING SOON ] /// /// [Tier 6] - GREECE QUANTITATIVE :[ 10^10^10^10^10^3000 - 10^10^10^10^10^10^3000 ]: CompactStar decided that Tier 6 is based on greek quantitative prefixes. In the greek language: mono / hapax means once, or one times dis means twice, or two times. tris means thrice, or three times. tetrakis means four times. and so on. You can check out the full list at the “numeral prefix” page. This “times” part makes sense by combining tier 5 roots as multipliers to tier 6. For example: “bexatetrakis” means “tetrakis” times “bexa” for me. Because tetrakis is 1012 in tier 5, and bexa is 200, the meaning can be interpreted as 200*1012. Like tier 5, he didn’t make tier 5 roots after the tier 6 milestones. That’s why I will make up additive roots for combining these tiers. Instead of using “e,” I decided to use a greek language of “plus” called “συν.” It’s pronounced as “syn” in English. It will be used for “-is” ending roots only. [subjoin - episynápto] Because there are 4 ending roots for tier 6 names, there will be a simple table below showing the normal/additive roots. Index Normal Additive Recommended 1 [ha]pax pa(ki/ka)- pax/paka 2 -dis d(u/i)sy(n/i/a)- dis/disya 3 -tris trisy(n/i/a)- tris/trisya 4+ -kis (ki)sy(n/i/a)- kis/sya [1] [ha] should be removed when it precedes with Tier 5. [2] You can choose any option in respective () roots. Here’s a table lookup for tier-6 -illions, because it is not difficult to fill up the roots beyond 20. Index Base Ones Ones (+10) Tens Hundreds 1 pax/paka hena hendeca hecatonta 2 dis/disya dya dodeca icos(a/i) diacos(a/i) 3 tris/trisya trya decatria triaconta triacos(a/i) 4 tetra decatetra tessaraconta tetracos(a/i) 5 penta decapenta penteconta pentacos(a/i) 6 hexa decahexa hexeconta hexacos(a/i) 7 hepta decahepta hebdomeconta heptacos(a/i) 8 octa decaocta ogdoeconta octacos(a/i) 9 enna decaennea eneneconta enneacos(a/i) 10 deca For naming tier 6: If x < 4, then only use the base names. Use the second option if tier 5 is before it. Do not use the tens column, if x mod 100 is less than 20. Instead, use the second “ones” column for roots between 10th and 20th. If there is (a/i) option, then use a if there isn’t a tier 6 root before you. Elsewise, use i. Finally, add -akis at the end. Use -asya instead if tier 5 is before it. That’s it! Now, here is an overview table for tier-5 -illions beyond hapaxillion. Number Tier 5 Index Tier 6 Index Name Alt. H5#1,000 1,000 1 hapaxillion H4#(103,000+1) 1,000+ 1+ hapaxe-hepillion H4#(2103,000) 1,000 meje-hapaxillion H5#1,001 1,001 hapaka-hepillion hapa’hepillion H5#1,002 1,002 hapaka-ottillion hapa’ottillion H5#1,010 1,010 hapaka-geopillion H5#1,100 1,100 hapaka-alphillion H5#2,000 2,000 1* ottapaxillion otig’paxillion H5#2,001 2,001 otig’paka-hepillion ottapaka-hepillion otig’pa’hepillion H5#3,000 3,000 neatapaxillion neg’paxillion H5#10,000 10,000 geopapaxillion geg’paxillion H5#11,000 11,000 gea’hepapaxillion geg’heg’paxillion geg’hepapaxillion H5#11,001 11,001 gea’heg’paka-hepillion H5#20,000 20,000 amosapaxillion amg’paxillion H5#100,000 100,000 alphapaxillion alg’paxillion H5#1,000,000 1,000,000 2 disillion H5#1,000,001 1,000,001 2+ disya-hepillion dusyn’hepillion H5#2,000,000 2,000,000 2* ottadisillion otig’disillion H5#2,000,001 2,000,001 otig’disya-hepillion H5#10,000,000 10,000,000 geopadisillion H5#100,000,000 100,000,000 alphadisillion H5#1,000,000,000 1,000,000,000 3 trisillion H5#1,000,000,001 1,000,000,001 3+ trisyi-hepillion trisyn’hepillion H5#1,000,000,000,000 1,000,000,000,000 4 tetrakisillion H5#1,000,000,000,001 1,000,000,000,001 4+ tetrasyi-hepillion tetrakisyn’hepillion Tier 6 Index Name H6#5 pentakisillion H6#6 hexakisillion H6#7 heptakisillion H6#8 octakisillion H6#9 ennakisillion H6#10 decakisillion H6#11 hendecakisillion H6#12 dodecakisillion H6#13 decatriakisillion H6#14 decatetrakisillion H6#15 decapentakisillion H6#20 icosakisillion H6#21 icosihenakisillion H6#22 icosidyakisillion H6#23 icositryakisillion H6#24 icositetrakisillion H6#25 icosipentakisillion H6#30 triacontakisillion H6#31 triacontahenakisillion H6#40 tesseracontakisillion H6#41 tesseracontahenakisillion H6#50 pentecontakisillion H6#60 hexecontakisillion H6#70 hebdomecontakisillion H6#80 ogdoecontakisillion H6#90 enenecontakisillion H6#100 hectatonakisillion H6#101 hectatonahenakisillion H6#110 hectatonadecakisillion H6#111 hectatonahendecakisillion H6#120 hectatonaicosakisillion H6#200 diacosakisillion H6#201 diacosihenakisillion H6#300 triacosakisillion H6#400 tetracosakisillion H6#500 pentacosakisillion H6#600 hexacosakisillion H6#700 heptacosakisillion H6#800 octacosakisillion H6#900 ennacosakisillion H6#990 ennacosienenecontakisillion H6#999 ennacosienenecontaennakisillion [OLD TIER 5 MULTIPLICATIVES] At this point, precedent roots are replaced, if directly followed by tier 6 then tier 5. Index Ones Tens Hundreds 1 heg’ geg’ alg’ 2 otig’ amg’ beg’ 3 neg’ hag’ gag’ 4 deg’ kyg’ deg’ 5 ung’ pig’ theg’ 6 eng’ sag’ iog’ 7 fig’ preg’ kag’ 8 syg’ nig’ lag’ 9 brog’ zog’ sig’ (example: otig’paka-ottillion = 2,002nd Tier 5) (example: sigma’nig’disya-hapaxillion = 980,001,000th Tier 5) We already entered the Tetrational class, surpassing Class 6! It would be the best stopping point for this document. The main part is over. Feel free to go somewhere, or click the links below for curiosity. Naming 10^^n numbers Extended Standard Sheet Overview DeepLineMadom’s Alternate MrRedShark77’s Continuation Aarex’s Googology “Onto the journey of endless stairs of googology.“ “Thy Eternal, Endless of Zillions.“ [[ SPECTRAL REALM ]] Welcome to the realm of tetrational! Exponents now feel close between 2 floors, thus aren't significant. We still feel like simpler formulas need special treatments from illions, hence low Tier 1. [Tier 7] - HEPTACHROMATIC :[ 10^10^10^10^10^10^3000 - 10^10^10^10^10^10^3,000,000 ]: [CHECK OUT NEW ILLIONS RIGHT] Redillion = 1,000-th tier 6 -illion. (That’s 1,000^^7*1,000!) (not by me, but it is found somewhere around 2020 - 2021) But I am not done yet. I need to combine it with tier 6 multipliers / additives to finish this. Tier 5 additives: Add “is” on the right; the option is the “sy” part! Tier 6 additives: Change “red” with “resi” or “redona,” and replace the base roots below. 1: enkapax 2: dyakis 3: tryakis Tier 6 multiplicatives: Just replace the final tier 6 roots with specific roots below: Index Ones Tens Hundreds 1 heke decise hecatise 2 dyke icoise diacise 3 tryke triacone triacise 4 tetrke tessaracone tetracise 5 penke pentecone pentacise 6 hexke hexecone hexacise 7 eptike hebdomecone heptacise 8 oceke ogdocone octacise 9 eneke enenecone enneacise No, there aren’t base roots for tier 6 multiplicatives. Tier 6 Index Name Tier 6 Index Name 1,000 redillion 2,000 dyke-redillion 1,000+ redis-hapaxillion 2,000+ dyke-redis-hapaxillion 1,000* ottaredillion 2,001 dyke-resi-ekapaxillion 1,001 resi-ekapaxillion 2,010 dyke-resi-decakisillion 1,001+ resi-ekapa-hepillion 3,000 tryke-redillion 1,002 resi-dyakisillion 4,000 tetrke-redillion 1,002+ resi-dyasyn-hepillion 5,000 penke-redillion 1,003 resi-tryakisillion 6,000 hexke-redillion 1,004 resi-tetrakisillion 7,000 epitke-redillion 1,005 resi-pentakisillion 8,000 oceke-redillion 1,010 resi-decakisillion 9,000 eneke-redillion 1,011 resi-hendecakisillion 10,000 decise-redillion 1,012 resi-dodecakisillion 11,000 decaheke-redillion 1,013 resi-decatriakisillion 12,000 decadyke-redillion 1,020 resi-icosakisillion 20,000 icoise-redillion 1,021 resi-icosahenakisillion 21,000 icosaheke-redillion 1,030 resi-triacontakisillion 30,000 triacone-redillion 1,100 resi-hectatonakisillion 31,000 triacontaheke-redillion Tier 6 Index Name 40,000 tesseracone-redillion 50,000 pentecone-redillion 60,000 hexecone-redillion 70,000 hebdomecone-redillion 80,000 ogdocone-redillion 90,000 enenecone-redillion 100,000 hectise-redillion 101,000 hectatonaheke-redillion 110,000 hectatonadecise-redillion 200,000 diacise-redillion 201,000 diacosiheke-redillion 300,000 triacise-redillion 400,000 tetracise-redillion 500,000 pentacise-redillion 600,000 hexacise-redillion 700,000 heptacise-redillion 800,000 octacise-redillion 900,000 ennacise-redillion 990,000 ennacosienenecone-redillion 999,000 ennacosienenecontaeneke-redillion [Tier 8] - … Due to the randomness, Cloudy176's faketest won't be included in Tiers 5+. [The Finale] - INFINITE HORIZON The plan for this chapter is to make an arbitrary customizing scheme with adjustable tiers. To do this, we need a new format covering 3 attributes, and 1 additional attribute: Tier. “Tier” means it is a layer from the power tower of 1,000s. We name Tiers as greek numeral prefixes but ending with e instead of a, except that Tier 7 is re, not hepte. Now, we are settling off from the superexponential range! Coming onto the normal attributes, we need a custom one. We observe “d” after “re,” so it is assigned as 1 in a tier coefficient. Thus, octedillion is 1st tier-8 illion, around H8#1. For 2nd tier-7 -illion, I would call this redualillion, making “dual” as coefficient 2 -affix. It feels like a mutant off from the start. Then, we can use the Greek number prefixes with additional “-as” ending. This completes the naming gap between spectral tiers. (ennal = 9x, pentaicosal = 25x) Here are tier 7 milestones. Tier 7 Index Name H1#7 redillion H2#7 redualillion H3#7 retrialillion H4#7 retetralillion H5#7 repentalillion H6#7 rehexalillion H7#7 reheptalillion H8#7 reoctalillion H9#7 re-ennealillion H10#7 redecalillion H11#7 rehendecalillion H12#7 redodecalillion H13#7 retridecalillion H14#7 retetradecalillion H15#7 repentadecalillion H16#7 rehexadecalillion H17#7 reheptadecalillion H18#7 reoctadecalillion H19#7 reennadecalillion H20#7 reicosalillion H21#7 rehenicosalillion H30#7 retriacontalillion H40#7 retetracontalillion H50#7 repentacontalillion H60#7 rehexacontalillion H70#7 reheptacontalillion H80#7 reoctacontalillion H90#7 re-enneacontalillion H100#7 rehectalillion For Tier 6 additives: we have “resi,” so how about we do the same? Just replace “al” with “asi” instead! Tier 6 Index Name H1,001#6 resi-enkapaxillion H1,000,001#6 reduesi-enkapaxillion H1,000,000,001#6 retriesi-enkapaxillion And now, the moment of “infinitum.” For tier-x multiplicatives, remove “l” from “al.” ([tier prefix][quantitative value x]+”a” = *x to Tier-[tier] value) For tier-x additives, replace “al” with “e” + [tier prefix] (“e”+[tier prefix][quantitative value x] = +x to Tier-[tier] value) If there’s “ed” instead of “al,” use “ae” + [tier prefix] If we add a root as a tier-n additive, but not at tier-n originally, add “e” + [tier-n prefix]. Phew, I’m now done climbing the infinite ladder! Value Name H1#7 redillion H103,000+1#5 redis-hepillion H103,000+1,000#5 redis-hapaxillion H1,001#6 resi-enkapaxillion H2,000#6 dyke-redillion H10,000#6 decise-redillion H100,000#6 hectise-redillion H2#7 redualillion H1,000,001#6 reduasi-enkapaxillion H10,000,000#6 decise-redualillion H3#7 retrialillion H10#7 redecalillion H100#7 rehectalillion H1#8 octedillion H103,000+1#6 octesi-enkapaxillion H103,000+1,000#6 octesi-redillion H1,001#7 octae-redillion H2,000#7 redua-octedillion H10,000#7 redeca-octedillion H100,000#7 rehecta-octedillion H2#8 octedualillion H1,000,001#7 octedue-redillion H10,000,000#7 redeca-octedualillion H3#8 octetrialillion H10#8 octedecalillion H100#8 octehectalillion H1#9 ennedillion H103,000+1#7 ennae-redillion H103,000+1,000#7 ennae-re-octedillion H1,001#8 ennae-octedillion H2,000#8 octedua-ennedillion H10,000#8 octedeca-ennedillion H100,000#8 octehecta-ennedillion H2#9 ennedualillion H1,000,001#8 ennedue-octedillion H10,000,000#8 octedeca-ennedualillion H3#9 ennetrialillion H10#9 ennedecalillion H100#9 ennehectalillion H1#10 decedillion H103,000+1#8 decae-octedillion H103,000+1,000#8 decae-octe-ennedillion H1,001#9 decae-ennedillion H2,000#9 ennedua-decedillion H10,000#9 ennedeca-decedillion H100,000#9 ennehecta-decedillion H2#10 decedualillion H1,000,001#9 decedue-ennedillion H10,000,000#9 ennedeca-decedualillion H3#10 decetrialillion H10#10 decedecalillion H100#10 decehectalillion And so on… Imagine the infinite tiers of illions! Now, the imagination came true. Mind-blowing! Lightning’s out! You can check out my other naming numbers: 10^^x numbers Hyper-E Numbers Extended Standard Sheet Overview DeepLineMadom’s Alternate MrRedShark77’s Continuation Trialogue = 10^10^10 Tetralogue = 10^10^10^10 Pentalogue = 10^10^10^10^10 .... Dekalogue = 10^^10 Hectalogue / Giggol = 10^^100 Chilialogue / Giggolchime = 10^^1000 Tier 7: Redillion Bitterswillion Vermillion Orangillion Goldillion Yellillion Limillion YGillion Greenillion Aquillion Cyanillion Tealillion Blueillion Indigillion (Maoillion) Purpillion Violillion Tramaoillion Tetramaoillion Pentamaoillion Hexamaoillion Heptmaoillion Octamaoillion Ennamaoillion Decamaoillion Magentillion Pinkillion Blackillion Whitillion Greyillion Silvillion Skinillion Brownillion Handillion Wigillion Switillion Orangitillion Tier 8: Erkillion Alejandrillion Bazillion Bethillion Bagillion Bridgillion Codillion Courtillion Devillion Duncillion Evillion Ezekillion Geofillion Gwenillion Harolillion Heathillion Jojolion Tier 9: Izzillion Justillion Katillion Leshawnillion Lindsillion Noahillion Owenillion Sadillion Sierrillion Trentillion Tylillion Derfillion Brittillion Vocillion Proudillion Trainillion Painillion Babillion Scillion Troptillion Bitillion Mitillion Koptillion Hiptillion Dayillion Ontillion Chirpillion Netillion Tier 10: Meakoowillion Wookipillion Coopillion Morefillion Royardillion Troyardillion Tetroyardillion Pentoyardillion Hexoyardillion Heptoyardillion Octoyardillion Ennoyardillion Hogillion Icosoyardillion Pentacontoyardillion Tier 11: Mercurillion Venusillion Earthillion Marsillion Jupitillion Saturnillion Uranillion Neptunillion Plutillion Sunillion Erisillion Dysnomillion Makemakillion Haumillion Gijebrillion Asebrillion Lunebrillion Fermebrillion Jovebrillion Solebrillion Betebrillion Glocebrillion Gaxebrillion Supebrillion Versebrillion Multebrillion Pyrebrillion Guntebrillion Kentrebrillion Onlebrillion Paptrebrillion Housebrillion Trongebrillion Batebrillion Handrebrillion Mastebrillion Gotebrillion Kortebrillion Nenkebrillion Janebrillion Unillion Deuxillion Troisillion Quatrillion Cinqillion Sixillion Sepillion Huitillion Neufillion Dixillion Onzillion Douzillion Treizillion Quatorzillion Quinzillion Seizillion Dixsepillion Dixhuitillion Dixneufillion Vingtillion Trenillion Quarantillion Cinquantillion Soixantillion Soixantedixillion illion Quatrevingtedixillion Cenillion Deuxecenillion Troisecenillion Quatrecenillion Cinqecenillion Sixecenillion Sepecenillion Huitecenillion Neufecenillion Milleillion Hugillion Bigillion Largillion Enormillion Superiorillion Nuclearillion Destructillion Tier 12: Doxillion Hevillillion Traminfillion Goillion Formalillionantillion Jovillillion Inphixillion Cillillion Fagillion Begillion Mineillion Robloxillion Gabillion Jillillion Hexillion Hextillion Seemillion Vagillion Vokillion Traktillion Satrillion Trafllion Centrillillion Rekillillillion Keitillion Formillion Hapillion Frannillion Moxantrajantramaoillion Krisillion MrTopFillion Dolphillion Thomillion Kaffwillion Ion Tier 13: Transillion Fakeillion Triktillion Amsterdillion Minionillion Triacosillion Tetracosillion Pentacosillion Hexacosillion Heptacosillion Octacosillion Ennacosillion Dumillion Gastroadillion Tromillion Omegillion Megamillion Hexttillion Xetillion Vockillion Yoctrikillion Multrillion Elongillion Houzillion Howillion Skyscraperillion Americillion Citillion Carillion Busillion Truckillion Planillion Helicoptrillion Tsarillion Slipillion Tramillion Franillion Jogillion Mocillion Moxillion Moxhamillion Hammerillion Malillion Contillion Dontillion Gontillion Fontillion Hontillion Kontillion Korillion Zerillion Killinillion Hogtillion Seperillion Gamillion Hadramillion Xandrillion Dentillion Bentillion Gazillion Squillion Bajillion Bazillion Jillion Zillion Haxillion Dakkillion Ikkillion Trakkillion Googolillion Oldillion Sexamaoianamdanabsolutedestructillion Hextramillion Heptillion Hopillion Daycarillion Schoolillion Collegillion Ayillion Summerillion Autumnillion Winterillion Springillion Hydroxillion Hotyillion Meccillion Xerillion Autillion Mansionillion Richillion Chinaillion Amazillion Farmillion Trulillion Bulkillion Containerillion Coachillion Idillion Fragmentillion Denptrillion Dentistillion Docterillion Torillion Tarillion Jerillion Detectillion Srirachillion Hamsterillion Tierillion Zackillion Kyrieillion Kattillion Royillion Jansenillion Jerlillion Lorelynillion Maggillion Janeillion Einstillion Kirbillion Eithillion Yocillion Yogillion Turillion Turbillion Oillillion Housalinatadenctahatramaoillion Cillion Honterillion Tier 14: Swimillion Censorillion Villillion Kaltillion Alfillion Rebelaxillion Kathillon Hillillion Fillion Sillion Jillion Zillion Tragicillion Allergicillion Glosillion Gillillion Gradillion Fertillion Daykillion Cillillion Meggillion Giggillion Killillillion Millillillion Gentillion Gintillion Frecklilllion Tractorillion Tortillion Tartillion Nukillion Tortillillion Pieillion Fryillion Xenoxenillion Quintrillion Hepttillion Sotrandillion Fugillion Fungusillion Boxillion Tragillion Alpillion Alkalillion Violinillion Guitarillion Pianillion Organillion Gunillion Paperillion Darkillion Papillion Hurtillion Jakillion Hexamaisadennisianadentamaoillion Exersisillion Trackillion Ickillion Dackillion Micillion Mousillion Dogillion Ramaxillion Ramaxonillion Grendillion Tiehillion Trillillion Pillillion Pillowillion Ice-Creamillion Dogfianillion Treeillion Woodillion Factrillion Sembillion Tamborillion Rimcktillion Aujillion Fijillion Vectrillion Dectillion Fyamillion Hexttillion Textillion Texillion Mexillion Mexicillion Frecklillion Jinillion Missillion Masillion Idrillion Freckillion Astrillion Jovendiramaoillion Gehillion Bukillion Ukrillion Xunillion Verillion Merrillion Aardillion Bravillion Charlillion Delphillion Primillion Obvillion Erkamillion Erillion Micramaoillion Heptanahadrillion Homillion Gonillion Heroillion Policillion Catillion Farmerillion Buildingillion Ckamnillion Damillion Novillion Novamosillion Bosnillion Hermillion Centrifugillion Dransillion Frecklillion Tria-gentillion Tetra-gentillion Penta-gentillion Hexa-gentillion Hepta-gentillion Octa-gentillion Enna-gentillion Triaillion Tetrillion Pentillion Hexxillion Hepptillion Occtillion Ennillion Deccillion Fermmillion Xendakultamultiamaoillion Sevillion Absillion Charlesillion Sectionillion Ikurtansadaurtanikillafreckillion Trakentramaillion Maillillion Jectillion Mectillion Fermentillion AumSumillion MrBeastillion Indianillion Lungillion Liverillion Heartillion Humanbodyillion Gorillaillion Elephantillion Rhinocerosillion Zebraillion Hamsterillion Caribouillion Dogillion Catillion TRexillion Triceratopsillion Crocodileillion Deinosuchusillion Flowerillion Treeillion Notoillion Ncvrillion Humnaillion Ultrakillion Tier 15: Frecklillion Jectillion Yocktillion Xonnillion Zeptrillion Heptrillion Hextrillion Sotillion Rimtillion Troctillion Hentillion Frankillion Itillion Mitchelillion Gogillion Gidgillion Chloeillion Adamillion Jodieillion Heptahapaxenultaquinquagintillion Xannillion Xendillion Globillion Firmillion Fumillion Ferillion Zerillion Millinillillion Tragillion Alfamartillion Malillion Gradillion Grahillion Bukuwillion Scrillion Quinquagintramillion Cendillion Kentillion Ventillion Violatillion Redsillion Legillion Atillion Roysillion Pebblillion Hectyillion Tirillion Ramillion Vigillion Classillion Astrillion Trudillion Harmonillion Octamafrandatechnoiadtyantylillion Villillion Classicalillion Buxillion Bulillion Penillion Pennyillion Denchardillion Benchardillion Yocillion Mantillion Barbillion Quecktosamasteanasotredagentillion Nonagillion Gillillion Fillinillion Killinillion Irillion Ibillion Sodrillion Sexeragentrantagstarastarharlianuatrucentanamillinillion Xenultillion Raxillion Ranaxillion Ranaxonillion Tetrisillion GTAViceCityillion Jocillion Vacillion Pentachillion Yectillion Yulillion Moldovillion Roanillion Transantanillion Xenillion Syrupillion Techillion Lenillion Mohillion Ukillion Proudillion Hondillion Centauillion Xenillion Towerillion Tiredillion Sakllion Togillion Trailillion Fuyillion Huyillion Dectillion Charlieillion Delphinillion Quarantillion Bukuwahillion Cosmeticillion Youtubillion Bankruptillion Violinillion Vigillion Trigillion Quadragillion Quinquagillion Sexagillion Septuagillion Octogillion Nonagillion Centrillion Millunadecamillion Grangillion Millibadratrahadabukuwahadatreeantarillion Absillion Agonillion Panickillion Heppillion Boogillion Boovillion Breachillion Hapexillion Yetillion Motherillion Gayillion Millillillillillillillion Barkillion Logillion Ultrillion Volcillion Infillion Gillillion Kansasillion Susillion Anusillion Thetapillion Novagillion Vagintillion Housajillion Gayajdahamokilacuillion Longdogonillion Ashortleashillion Tier 16: Javillion Freyillion Ukrainilion Hetillion Killinillinillion Maxillion Thowillion Millionillion Ownillion Kerillion Otterillion Macillion Mcdonaldsillion Vaginillion Hempillion Hectarillion Giggillion Treehillion Rillillion Zerkillion Mothillion Vajillion Lahillion Samillion Mallgillion Houseillion Gringaintanyatagentillion Gringillion Oktillion Otillion Playillion Daycillion Grassillion Assillion Xillion Grasshopperillion Hextamillion Hextamillion Sosillion Seoulillion NYCillion LAillion Knowillion Vectillion Cosillion Kisillion Trivillion Gantillion Gangstrillion Gangsterillion Gagillion Missolillion Mossolplexillion Jakeillion Atticillion Lightillion Aarhillion Meakillion Arbillion Icillion Polillion Vifillion Dectillion Gecillion Centyillion Dubillion Vecxillion Becxillion Mecxillion Hecxillion Gentrillion Generillion Vocalillion Obillion Violillillion Nomillion Oilillion Dachillion Xenoxenoxillion Cirillion Cerillion Millingillion Vogillion Trammyillion Mectrillion Ducillion Heroinillion Dakanadranameganamegillion Giftillion Logueillion Trigillion Tetragonillion Pentagonillion Hexagonillion Heptagonillion Octagonillion Ennagonillion Hecillion Kintrillion Mecgillion Otherillion Maxtillion Deckillion Treenillion Quinvigeranillion Giggolillion Herbavrollion Tienillion Hextamillion Pentirillion Pyramidillion Deflillion Throwillion Berillion Tardillion Erillion Pepillion, Peptrillion Oxillion Oxfordillion Kermillion Billimafranaxeantarahapaxennillion Xenoxenoxenillion Denchillion Denctillion Dragonillion Limixillion Radillion Morbillion Megamindezearazeusillion Invillion Januarillion Trigrilantrillion Hillillillion Limousinillion Vagin-Verillion Satillion Sotillion Tier 17: Pentillion Pent-Million Pent-Billion Pent-Trillion Pent-Decillion Pent-Vigintillion Pent-Googol Pent-Quadragintillion Pent-Centillion Pent-Millillion Pent-Myrillion Pent-Micrillion Pent-Googolplex Pent-Hectillion Pent-Killillion Pent-Ikillion Pent-Trakillion Pent-Multillion Pent-Hepillion Pent-Augillion Pent-Hapaxillion (also known as Italillion) Pent-Disillion Pent-Trisillion Pent-Tramaoillion Pent-Decamaoillion Pent-Sotillion Pent-Vagin-Verillion Pent-Charlillion Pent-Tramillion Limitillion Fujillion Hujillion Gujillion Bujillion Dujillion Mujillion Nujillion Limmillion Lyrillion Myrillionmyrillion Moonillion Tiredillion Herillion Erckillion Megacillion Googolplexillion Citillion Teslillon Unteslillion Duoteslillion Triteslillion Quattuorteslillion Quinteslillion Sexteslillion Septemteslillion Octiteslillion Novemteslillion Miragillion Rayillion Jocillion Idyeavocalillion Lillion Trillillillillion Thetapaxillion Xeroxillion Sexillion Vexillion Ravillion Villagerillion Chickillion Pigillion Technoblillion Elillion Vagin-Tramaoillion Verillion Aaronillion Arghillion Brillion Vesillion Gevillion Pingillion Ngerillion Nigillion Molillion Tranquillion Pentyeatrartaratreagentatrigentillion Chavillion Oblivillion Infinitillion Finchillion Triacillion Tetracillion Pentacillion Hexacillion Heptacillion Ocktacillion Ennacillion Dekillion Ickosillion Triacontacillion Tetracontacillion Pentacontacillion Hexacontacillion Heptacontacillion Octacontacillion Ennacontacillion Hectacillion Chillacillion Myriacillion Taxillion Trucillion Trusillion Rustillion Mistillion Kallillion Harlillion Funkillion Dojillion Platrillion Illionillion Ilillion Lovillion Garrayantaragayillion Afragarearafillion Absolutillion Sextratailllion Grisillion Strillion Ikyokillillion Nonecxaxenultillion Tiramisillion Tridekillion Trafillion Ricillion Tier 18: Freakillion Outsidillion Orckillion Googillion Gagillion Gobillion Oblivillillion Tracyillion Freudillion Eudillion Udillion Nobillion Conanillion Doraemillion Novillillion Hexahedrillion Novemdreagarearraradrattayectagintillion Cintillion Ducintillion Trucintillion Quadringintillion Quingintillion Sescintillion Septingintillion Octingintillion Nongintillion Tranquintanovemgaentaedaorantacxentutrambutragayillion Ficnillion Demillion Mellillion Plantillion Animalillion Bacterillion Durillion Danchillion Subillion Ganchillion Ranchillion Rantillion Banchillion Fanchillion Kanchillion Knowillion Swillion Demillion Octataratreagetratratrertrakantratramaintagentillion Delphinillion Walillion Owillion Daneillion Lillintrantamillion Bentlillion Grahillion Bukuwahantramillion Piccillion Prillion Drantillion Kyrillion Kyr-Million Kyr-Billion Kyr-Decillion Kyr-Vigintillion Kyr-Centillion Kyr-Millinillion (Millillion) Kyr-Myrillion Kyr-Micrillion Kyr-Triacontillion Kyr-Hectillion Kyr-Megillion Kyr-Trakillion Kyr-Disillion Kyr-Tramaoillion Kyr-Decamaoillion Kyr-Erkillion Kyr-Kyrillion Dukyrillion Dukuwillion Astlillion Dohectakisillion Chillakisillion Myriakisillion Audillion Vunduktillion Untillion Bentrizillion Troktillion Sotrillion Rimktillion Quektillion Ocktillion Googolnoninplexithrongillion Tollillion Grillion Ennayvunduktsarbombrabtratrakengentillion Enviousillion Encikillion Illishoikillion Mercktaretrathrongethreatreantagayolintramillion Nonexillillion Goxillion Cosillion Garrel-million-trucentillion Gecrtrillion Aardillion Myriamaoillion Chinillion Premiumillion Satillion Nonecxentramaoillion Centytramillion Tier 19: Charlillilllillillion Orcantardillion Foxxillion Millinamaoilion Airlarillion Megamaoillion Sextraudwanuratyliatramaoillion Nancillion Unitedillion Unitillion Freshillion Phillion Goxxillion Ovillion Nonexenultraeuntraillion Conanillion Nobellillion Safeillion Giornoillion Pentaconacontillion Hectamatrillion Cevillion Altaccfillion Jinglillion Grillillion Parillion Mertillion Boovillion Boogollillion Dakotillion Ictreayttartruetranatwakillion Shapillion Hooverrillion Daarrillion Dhaarmannillion Dhenillion Whillion Huntrillion Hutaftraillion Atmillion Gijamsntatillion Darrellillion Barrellillillion Foxintarillion Torendillion Metallillion Omnivorousillion Sandpillion Maccillion Farrillion Harxillion Sortrillion Gogillion Giigillion Terramillion Tesserillion Ggtatartueillion Guweuiequiqyillion Gruelillion Villion Deckillion Unveilillion Gogologueillion Plexillion Descaonudustuntratambillion Plexidingillion Chimillion Berjillion Ferjillion Terjillion Gerjillion Derjillion Nerjillion Firjillion Gingivitillion Ingillion Oncillion Gaxenillion Xeroxeroxeroxoillion Megrillion Gigrillion Terrillion Pettillion Exetillion Zetillion Yotillion Xen-illion Drakillion Irakillion Trokillion Ponininijillion Mijillion Ferillion Asillion Contamaoillion Pentacontrilatrareqtreatrwacxentaramaatranatatracontillion Contyillion Hassolugulillion Sextramajuewyillion Cxillion Environmenillion Housallillion Ratrillion Sutillion Sugillion Sugarrillion Farmillion Hextrantacontillion Coneillion Cylindrillion HexagonalpriGogsmillion Cubillion Spherillion Supercalifragilsticexpialidociousillion Teachillion Cityillion Hapaxillillion Disillillion (Isillion) Jurillon Ovillion Hoxillion Hoverrillion Narrillion Hapacksillion Itallillion Australillion Canadillion Moldillillion Turkmillion Hillion Bhutillion Wuntillion Wanillion Sagannillion Annuntiraillion Naenillion Tier 20: Trektillion Untrektillion Duotrektillion Tretrektillion Quattuortrektillion Quintrektillion Sextrektillion Septingtrektillion Octotrektillion Novemtrektillion Decitrektillion Centitrektillion Ducentitrektillion Trucentitrektillion Quadringentitrektillion Quinquagentitrektillion Sexentitrektillion Septingentitrektillion Octingentitrektillion Nongentitrektillion Millitrektillion Decimillitrektillion Centimillitrektillion Micritrektillion Nanitrektillion Picitrektillion Femtitrektillion Attetrektillion Zeptitrektillion Yoctitrektillion Xonitrektillion Vecitrektillion Mecitrektillio{d}xIcositrektillion Triacontitrektillion Tetricontitrektillion Hectitrektillion Killitrektillion Megtrektillion Dutrektillion Trutrektillion Tetretrektillion Pentetrektillion Hexetrektillion Heptetrektillion Octetrektillion Ennetrektillion Dekatrektillion Gorillion Horserillion Giggillion Jamesillion Sotredraaoillion Doedillion Trakkanayillion Yillion Holillion Bolillion Galillion Myriakisillion Hextramolillion Mejantillion Aquaillion Aidillion Trentonillion Tentillion Gaytillion Hexttextillion Mextillion Pentaamaysyeayeweuwantaranaatcontillion Dark-Trillion Ogillion Verserillion Demillion Gexillion Vonillion Burgerillion Hamburgerillion Unvillion Gloxillion Gaexillion Absillion Transeranctahectrakiasexilion Pentacthulillion Goghoustillion Clacillion Astronomillion Goxygennillion Elonmuskillion Blueyillion Tellillion Goxantranaterillion Lineatrillion Empillion Gigillillillion Thantillion Housamaillion Xillinatmillion Atmosphillion Xevillion Inillion Aumstillion Grocillion Goxianillion Hossillion Mansillion Transfillion Roptillion Tropicillion Burundillion Lesothillion Malawillion Spainillion Americaillion Unitedkindomillion Francillion Québecillion Ontarillion Albertillion Manitobillion PEillion Novascotiaillion Newfoundlandillion BCillion Saskatchewanillion Nunavutillion Philippillion Tatillion Moshillion Maggillion Antillion Robiillion Dionillion Kurtillion Fumillion Xillion Unveilillion Gozzillion Goexillion Meakillion Wookillion Copillion Royillion Troyillion Tetroyillion Pentoyillion Hexoyillion Heptoyillion Octoyillion Ennoyillion Dekoyillion Icosoyillion Pentacontoyillion Hectoyillion Tramaoillillillion Pentamiaduwydwenxyiryiwillion Xyillion Shortdogonillion Alongleashillion Tier 21: Voxillion Duhillion Uhandyillion Atillion Miccillion Nannillion Piccillion Femmillion Atttillion Zepillion Yotillion Xovillion Vekillion Mekillion Duekillion Icokillion Kusnsillion Kunguklillion Kungillion Augillillion Hexamaoudieudwhisgyxwiygyuxgdiygddywgdxillion Xillaaishaysfyuqfwwfzgargoogolapiwdsumdxiuesmghedillion Hyfqillion Dgillion Etayusillion Drsdtsuwmugnwuxndefgyenxashillion Ysnwmgydguyegudmgeuygfmuryillion Hgyubsantillion Franciajuuezwmymgwuhfdengufnguyndneunxwugdnfeyneetfeygfillion \int \frac{\pi \sin\left(\sqrt{x}\right) \, \mathrm{e}^{\sqrt{x}}}{\sqrt{x}} \, \mathrm{d}x: Bexaillion Trexaillion Tetrexaillion Pentexaillion Hexexaillion Heptexaillion Octexaillion Ennexaillion Familyillion Babyillion Boyillion Girlillion Manillion Womanillion Dohillion Ohamillion Japanillion Freewillion Teptillion Violillillion Fillion Erkamillion Bitterillion Swillion Orillion Lillion Aqillion Blillion Indgillion Pupillion Villion Tamillion Rillion Dexiajyioillion Oioioillion Hapakillion Hartillion Meanillion Woopillion Cpillion Moreillion Ictillion Noctosillion Nonosillion Golvillion Vaxillion Hexialillion Rkillion Killillillion Vecekillillillion Hectekillillillion Megamegillion Micryamoillion Partillion Blueintaraiuquaajnaahsuauasganmanuaardillion Grinchilllion Santillion Holillion Buctillion Ogdoeillion Loanillion Gugillion Bridillion Lxeoliauaasmgsasgatrigintillion Obviousillion Moserillion Tier 23: Goisillion Gyitillion Itarillion Blilamaoillion Centurionillion Blasphemorgulusillion Treethreeillion Thothillion Tetillion Hesaihsuaillion Royardimieaaiuillion Bluentamillion Gahulaguajiliahuagfillion Biwlwlion Nillion Nonillillillion Millinillinillinillion Kajillion Dajillion Trajillion Tajillion Pajillion Exajillion Zajillion Yajillion Najillion Dakajillion Hotajillion Meghillion Gighillion Astrillion Luntoxamillion Jrettiamillion Gixillion Quinsaayillion Gfillion Hillion Hozoxillion Boxillion Lugillion Missolillion Blasphemillion Centurillion Gongillion Giaoillionajillion Giaillion Aillion Aug-Maxcillion Xcillion Meakoowardillion Wookipardillion Royardyardillion Yardillion Tramillion Xillinillion Millinillillinillinillion Yillillion Fartillion Shinamillion Mantrillion Binocsillion Kittillion Drillion Handriamaquadrillion Goramillion Xianillion Gulillion Goaniamtillion Matillion Mathematicsillion Sciencillion Englilion Filipillion Homeroomillion Apillion Myillion Myopiaillion Phobillion Ganemillion Hogamillion Hovamillion Hoovillion Hoxillion Fbillion Polillion Tolillion Thorillion Supermillion Batmillion Wondillion Willoughbillion Timillion Perfectillion Hoxiuisaosuaahgsdgsyillion Upillion Downillion Loudillion Duhamaillion Ybaillion Libtillion Lubillion Gexillion Goxiasolillion Gasanxillion Crashillion Hyamillion Ammillion Ohajnillion Oniamillion Xillinaillion Hardillion Fortamillion Firmillion Hippillion Vandalisillion Edillion Johnillion Georgillion Geraldillion Cooperillion Haroldillion Derpillion Goalillion Gpillion Pillillillion Ixillion Tramillillion Uabtamillion Ubtillion Ugtillion Uhtillion Untiamillion Ynatillion Dnalillillion Buamayillion Hamillion Hummerillion Toyotillion Decorillion Blueintramaoaillion Baxisoillion Baxillion Oskillion Haosmillion Smaillion Smashillion Hdillion Hdyillion Darrillion Garraillion Absolutelillion Gintamillion Ungentillion Dumnaillion Planyillion Tier 24: Goanaillion Yillion Yamillion Funtamillion Luntamillion Gisillion Glosillion Italyiamillion Italamillion Hoexillion Hamtillion Xillinillion Charillion Delphamillion Intamillion Untillillion Bentrizillillion Troktillillion Sotrillillion Rimktillillion Quektillillion Pektillillion Ocktillillion Vedillion Enosillion Heneosillion Cantaramialalioaannaillion Hexaamillion Gexillion Losillion Greatillion Big-Hossillion Goanimatillion Tramillillillion Tramaoillillillion Dakultarelillion Harellillion Dossillion Moasillion Ossillion Novellillion Decxillion Goxialillion Tranillion Hgaillion Oggillion Mikillion MekillIon Bekillion Trekillion Tetrekillion Pentekillion Hexekillion Heptekillion Octekillion Ennekillion Decekillion Icosekillion Triacontekillion Tetracontekillion Pentacontekillion Hexacontekillion Fintillion Hogamontamantarmackillion Idxillion Joctillion Kosillion Jintramrillion Killillillillillillillion Gargillion Harlillion Govillion Gohillion Buhillion Biotillion Nedillion Gitterswillion Borangillion Holdillion Lellillion Bimillion Leenillion Uquillion Glueillion Vindigillion Burpillion Volillion Yramaoillion Petramaoillion Hentamaoillion Gexamaoillion Jeptmaoillion Poctamaoillion Mennamaoillion Orkillion Blalejandrillion Mizzillion Goahillion Hentamillion Pexillion Gorillion Garillion Harillion Blasphillion Gogoltillion Gexamillion Examaoillion Zettamaoillion Doxamillion Hostamiaauahyaratyfatyafytfauafacillion Fossillion Kostillion Kormillion Kyrieillion Hyrillion Byrillion Fyrillion Gyramillion Augillillillillion Indonesillion Goramillion Dormamillion Xillanillion Mottamillion Kuhillion Kungulillion Gormilllillillillion Axillion Edillion Yoctacontillion Noachillion Champillion Pillinamillion Bintillion Gorikkillion Hortamillion Totiamillion Troktamillion Grandmothillion Grandpillion Pintillintamaoillion Hoanillion Oxamaoillion Yottamaoillion Xennamaoillion Dakamaoillion Ikamaoillion Trakamaoillion Tekamaoillion Pekamaoillion Exakamaoillion Zakamaoillion Yokamaoillion Nekamaoillion Entamillion Glizzyillion Duhamillion Glocillillion Gertamillion Merillion Berillion Trerillion Tetrerillion Penterillion Hexerillion Hepterillion Octerillion Ennerillion Dekerillion Tier 25: Giotillion Animillion Kolillion Dolillion Trolillion Tolillion Polillion Exolillion Zolillion Yolillion Xolilion Groanillion Hortamillion Groacillion Yocxillion Superfragillion Hormillion Rorillion Jocyillion Luckillion Dekaxillion Kaxillion Taxillion Buntamillion Gocillion Hapaxxillion Dissillion Trisquillion Joerillion Joruntillion Toxillion Godillion Cenillion Blasillion Gongulillion Greatillion Gwenamillion Kisillion Hisillion Bisillion Lisillion Tisillion Quillillion Hoeckillion Ckamnamillion Killinamillion Namillion Buamillion Yramillion Petramillion Nentamilion Aexamillion Eptamillion Poctamillion Nennamillion Gillillion Kortanaishuagyscudysifsifgsdysdfgfnuudngdillion Dralillion Kvillion Fvfvfvillion Fvillion Kistmillion Tmanatamaillion Tmaoilnatillion Tillillillillillillillion Concrillion Killinaminataillion Matillion Mortillion Stopillion Goillion Xirtamillion Urtillion Ouchillion Horta-Millinillinillion Yoctamillion Kalamaoillion Kolamillion Bolillion Trolilion Tetrolillion Pentolillion Hexolillion Heptolillion Octolillion Ennolillion Dekolillion Icosolillion Triantolillion Hectolillion Duaardillion Truaardillion Tertruaardillion Pentuaardillion Hexuaardillion Heptuaardillion Octuaardilion Ennuardillion Dekuaardillion Bolillianthillion Mirrillion Raxamillion Noobillon Glockillion Vulillion Tetamillion Tamillion Mejamaoillion Gijamaoillion Astamaoillion Lunamaoillion Fermamaoillion Jovamaoillion Solamaoillion Betamaoillion Glocamaoillion Gaxamaoillion Supermaoillion Versamaoillion Multamaoillion Nonecxenultamaoillion Hoerillion Ertillion Erfillion Wtgamaoillion Tvillion Goaushduqxtillion Gnsydwywgndyillion Yuiwdmdiuillion Jdywjuyillion Ihewygeuyengdefguyeillion Metamaoillion Xevamaoillion Idixamaoillion Hepamaoillion Ottamaoillion Neatamaoillion Goatcillion Troktosauntamillion Stramsotrillion Truktillion Setrillion Romktillion Guektillion Vektillion Pocktillion Ovedillion Novillinamillion Hortillion Dubravillion Trubravillion Tetrubravillion Pentubravillion Hexubravillion Heptubravillion Octubravillion Ennubravillion Dekubravillion Hostamillion Blakillion Cocillion Vettamillion Burjillion Pyramidillion Mountainillion Dahillion Hottillion Deatamaoillion Unatamaoillion Entamaoillion Fitamaoillion Sytamaoillion Brontamaoillion Geopamaoillion Alphamaoillion Hapaxamaoillion Decakisamaoillion Redamaoillion Tramaomaoillion Tetramaomaoillion Pentamaomaoillion Hexamaomaoillion Heptmaomaoillion Octamaomaoillion Ennamaomaoillion Decamaomaoillion Ingamillillion Erkamaoillion Izzamaoillion Meakoowamaoillion Atomamaoillion Humaoillion Staramaoillion Alphamaoillion Duhamaoillion Superfragilicexpialidociousamaoillion Sicillion Tier 26: Houxillion Inphillion Gocillion Hactillion Kallillion Medillion Gishillion Kurillion Crashamaoillion Tramamaoillion Aardamaoillion Exillillion Xintillion Xertamillion Xeenillion Tennillion Tractillion Bockillion Dramillion Astmillion Thantoxillion Ohsuhdgillion Gusauduiwesgudwnillion Sopillion Hepillillion Zetiantillion Tetramamaoillion Xillamaoillion Bravamaoillion Charlamaoillion Delphamaoillion Primamaoillion Obvamaoillion Ducharlillion Trucharlillion Tetrucharlillion Pentucharlillion Hexucharlillion Heptucharlillion Octucharlillion Ennucharlillion Dekucharlillion Icosucharlillion Untamaoaintamillillion Uillion Unsuctillion Duoillion Trillinaillion Dumintillion Sexillion Doxasillion Goxilnyillion Ontamaointillion Maoillinillion Pintillion Tetramaoillillion Hoxamilion Burjillillion Rotillion Tier 27: Icosillinamaoillillillion Illinamillion Namamaillion Royillinatillion Nadfillion Fillion Gortamillion Redsyillion Octacontillinamaoillintamillion Bliuillion Iuiuillion Lucillion Johnillion Jovannillion Gocillion Glaxillion Sthiamillion Bintillion Jinikillion Hoxamaoaianayatamaaoaaiaayajauayakaajayahananantillion Voccillion Tri-billion Hoxrillion Finamillion Finalillion ( Not the Lastillion ) Dakfinalillion Hotfinalillion Uauillion Fvfvillillion Fzgillion Kiltillion Horammillion Grannmamillion Pentamaoillillion Dudelphillion Trudelphillion Tetrudelphillion Pentudelphillion Hexudelphillion Heptudelphillion Octudelphillion Ennudelphillion Dekudelphillion Killinamillion Hamillion Ozzillion Minecrillion Roblillion Hoxsjwhdquxgillion Gdillion Groundillion Osillion Klunatikillion Ayasmarillion Mukbangillion Bridgesillion Tteybokillion Bochillion Bochahendatanushotoridakillion Hoxtillion Aeroplillion Ikultamaakosnxdysanunscillion Tranusxsgxhedcusybdufdsydfsbtssfdaatrainillion Superfragilcillion Gowillillion Klunatikillion Hoxiantillion Duprimillion Truprimillion Tetruprimillion Pentuprimillion Hexuprimillion Heptuprimillion Octuprimillion Ennuprimillion Dekuprimillion Trillinamillion Hoazillion Ghsaxdydgdwuyagusillion Croshillion Croatillion Bosnillion Montenegrillion Kosovillion Albanillion Greecillion Ocdsajovamillion Ogdillion Ooilahillion Tier 28: Hoxddillion Jostamillion Joplillion Roskanillion Jovillinillion Hociclillion Cicillion Rontillion Rostillion Kostillion Morrillion Jokillion Rhinillion Lionillion Tigillion Monkillion Animalillion Jodwillion Hjdixoillion Oixwhufillion Ugsguqwnguwikameakoowillion Kooilion Kookoowillion Meakoowillinillion Hiwikluntillion Destillion Hohohillion Hidsalillion Hoxillanillion Xohillion Xevantadkuntamillion Dkamillion Djamnillion Africillion Jocillion Joswuillion Hortiamillion Kistillion Askillion Hostillillion Planetillillillion Josillion Fosillion Fisyillion Flexillion Fiillion Illion Hoxislilaswuiqsduyillion Killamillion Korillion Jostiamillion Howillion Kostamillion Bismillion Hostmaillion Aillion Airillion Oxillion Ottamaoillillion Extramaoillion Hypermaoillion Hyperalillion Mosilamoillion Oimsillion Nosconillion Duobvillion Truobvillion Tetruobvillion Pentuobvillion Hexuobvillion Heptuobvillion Octuobvillion Ennuobvillion Dekuobvillion Hodamillion Modillion Dkamianmillion Denpentajupentacontoyardillion Aardillillion Bravillillion Charlillillion Delphillillion Primillillion Obvillillion Delphinillion Tarliamillion Liamillion Johamillion Johnnyillion Johnyillion Missolplexillion Admillion Kosiolsmillion Rosillion Kisillion Lillillillion Millillillillillillillion Oblivionillion Utteroblivionillion Infinityillion Omegaillion Epsilonillion Zetaillion Etaillion Phillion Gammaillion Churchillion Mahlillion Weakillion Absolutillion (Absolute Infinityillion) Everyillion Growthillion Endlessillion The-Endillion Voidillion Endillion Stoppingillion Zerillillillion Bondazozillion Tier 29: Gaggolillion Triaillion Kiallmaillion Gigolillion Emperollillion Hoajillion Hozuoillion Moxiamillion Rosmillion Hlozillion Hozizoalaiosjillion Jzozoillion Iaizhziuillion KillinMaoillion Doctillion Millionillion Killillionillion Aardillionillion Joxiamillamnqjsaiqxrhuieanschmittypleasehelpmeillion Jociamtamtrigintramaoillion Bravamaoillion Novillinillion Tier 30: Lelillion Adesresdertillion Dogillion Catillion Horsillion Hospitillion Illionillion Novillion Conditioillion Lillillion Berillion Scgillion Wasillion Ededededededededillion Grangolillion Googologyillion Doogillion Filillion Novillinillinillion Novillinillionillion Sescinyillion Beartillion Erillion Vcillion Trigintillillion Adressillion Eightillion Nineillion Tenthillion Hundredillion *%illion Zooillion Thousandillion Examples using the "H-illion function and -illion name (e.g. 1 quattortrigintillion, 1 micrekilatrecillion) format": H34 = H34#1 = H(1,000^34)#0 = 1,000*1,000^34 = 10^105 = 1 quattortrigintillion H2,013#2 = H(1,000^2,013)#1 = 1,000^(1,000^2,013+1) = 10^(3*10^6,039+3) = 1 micrekillatrecillion Finalillion = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{243}} } } } } } } }+3} Lastillion = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{273}} } } } } } } }+3} Endillion = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{303}} } } } } } } }+3} Limitillion (last of -illions in the list of googologisms in Googology Wiki) = 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times 10^{3\times10^{3003}} } } } } } } }+3} Use "quattor" for 4-illions (e.g. quattordecillion, quattorvigintillion, etc.) Use "trescentillion" for 10^312. Use the updated tier 5 -illions: Tier 5: Hepillion Ottillion Neatillion Deatillion Unatillion Entillion Fitillion Sytillion Brontillion Geopillion Geopahepillion Geopottillion Geopaneatillion Geopadeatillion Geopaunatillion Geopaentillion Geopafitillion Geopasytillion Geopabrontillion Amosillion Haprillion Kyrillion Pijillion Saganillion Pectrillion Nisabillion Zotzillion Alphillion Bexillion Gammillion Deltillion Thetillion Iotillion Kappillion Lambdillion Sigmillion Omegillion Hepillion = H1#5 Ottillion = H2#5 Ne’atillion / Negillion = H3#5 De’atillion / Degillion = H4#5 Un’atillion / Ungillion = H5#5 Entillion = H6#5 Fitillion = H7#5 Sytillion = H8#5 Brontillion = H9#5 Geopillion = H10#5 Geopahepillion (alt: Geo’hepillion) = H11#5 … Amosillion = H20#5 … Haprillion = H30#5 Kyrillion (also; spelled Pigillion) = H40#5 Pijillion = H50#5 Saganillion (alt: Sagillion) = H60#5 Pectrillion (alt: Prykillion) = H70#5 Nisabillion = H80#5 Zotzillion = H90#5 … Alphillion = H100#5 Bexillion / Bethillion = H200#5 Gammillion = H300#5 Deltillion = H400#5 Thetillion = H500#5 Iotillion = H600#5 Kappillion = H700#5 Lambdillion = H800#5 Sigmillion = H900#5 … Hapaxillion / Omegillion = H1000#5 = H1#6 Use the updated tier 7 -illions: Redillion = H1#7 Orangillion = H2#7 Yellillion / Yellowillion = H3#7 Chartreusillion = H4#7 Limillion = H5#7 Greenillion = H6#7 Aquillion = H7#7 Bluillion = H8#7 Purplillion = H9#7 Pinkillion = H10#7 Redpinkillion = H11#7 … Blackillion = H20#7 … Blackredillion = H21#7 … Whitillion = H30#7 Maroonillion = H40#7 Turquoisillion = H50#7 Brownillion = H60#7 Cereleainillion = H70#7 Azurillion = H80#7 Beigillion = H90#7 Violetillion = H100#7 … Fuchsillion = H200#7 Flaxillion = H300#7 Dogillion = H400#7 Hogillion = H500#7 Dimillion = H600#7 Indigillion = H700#7 Magentillion = H800#7 Grayillion / Greyillion = H900#7 … Hydrillion / Milkillion = H1000#7 = H1#8 Use the updated tier 8 -illions: Hydrillion / Milkillion = H1000#7 = H1#8 Helillion = H2#8 Lithillion = H3#8 Berylillion = H4#8 Boronillion = H5#8 Carbonillion = H6#8 Nitrogillion / Nitrogenillion = H7#8 Oxygillion / Oxygenillion = H8#8 Flourillion = H9#8 Neonillion = H10#8 Neo’hydrillion / Noehydro-illion = H11#8 … Sodillion = H20#8 Magnillion = H30#8 Aluminillion = H40#8 Silicillion = H50#8 Phosphorillion = H60#8 Sulphillion = H70#8 Chlorillion - H80#8 Argillion = H90#8 … Argo’flourillion = H99#8 Potassillion = H100#8 Calcillion = H200#8 Scandillion = H300#8 Titanillion = H400#8 Vanadillion = H500#8 Chromillion = H600#8 Manganillion = H700#8 Ironillion = H800#8 Cobaltillion = H900#8 … Monogonillion / Einillion / Nickelillion = H1000#8 = H1#9 Use the updated tier 4 -illions: Pyrillion = H15#4 Guntillion = H16#4 Kentrillion = H17#4 Onlillion / Omnivillion = H18#4 Paptrillion / Pastry’illion = H19#4 Housillion = H20#4 Housalillion (aka: “Horsillion”) = H21#4 .. Trongillion = H30#4 Batillion = H40#4 Handrillion = H50#4 Mastillion = H60#4 Gotillion = H70#4 Kortillion = H80#4 Nenkillion = H90#4 … Janillion = H100#4 Manillion = H200#4 Ganillion = H300#4 Astanillion = H400#4 Lanillion = H500#4 Fanillion = H600#4 Jovanillion = H700#4 Sanillion = H800#4 Banillion = H900#4 … Hepillion = H1000#4 = H1#5 Killillion = 10^(3x10^3000+3) … Killo-decillion = 10^(3x10^3000+33) etc. Ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyrillion = 10↑↑10↑↑10↑↑10↑↑2000001 UNOCF (Username’s OCF) Ordinal LNGI beyond psi_0(N), a number greater than psi_0(M): ψ_0(Ν) ψ_0(Ν+1) ψ_0(Ν2) ... ψ_0(Ν^2) ψ_0(Ν^ω) ... ψ(Ν^^ω) = ψ_0(ε_{Ν+1}) = ψ(Ν_2)) ... ψ(Ν_ω) ... ψ(Ν_Ν_Ν_...) = ψ(Ν(1,0)) … Beyond Absolute Infinity: Absolute Infinity + 1 … Transfinity Superfinity Omegafinity Glitchfinity … Pink Sheep Number (SN(10^100)) Purple Sheep Number … Blue Sheep Number … Green Sheep Number … Yellow Sheep Number Orange Sheep Number Red Sheep Number Gold Sheep Number Silver Sheep Number … Absolutely Everything = ΩxΩ Absolutely Godvrything = ΩxXxΩ … Absolute Omega = Ω(ω) … Absolute True End … The End of Numbers … ABSOLUTE BEYOND TRUE END = Ω|Ω|Ω Viswanathan's function = Viswanthan(n) Vsauce’s function = Vsauce(n) - bigger than Graham’s Number Bashicu Matrix System Examples: \begin{pmatrix} 0 & 1 & 2 & 3 & 3\\ 0 & 1 & 2 & 3 & 2 \end{pmatrix} = (0,0)(1,1)(2,2)(3,3)(3,2) = \psi(\psi_1(\Omega_2)) = \psi_0(\psi_1(\psi_2(\psi_3(0)+\psi_2(0)))) = \text{or}\ >\ \psi_0(\Gamma_{\Omega_2+1}) \begin{array}{ll} (0) &=& 1 \\ (0)(0) &=& 2 \\ (0)(0)(0) &=& 3 \\ (0)(1) &=& \omega \\ (0)(1)(0) &=& \omega+1 \\ (0)(1)(0)(0) &=& \omega+2 \\ (0)(1)(0)(1) &=& \omega2 \\ (0)(1)(1) &=& \omega^2 \\ (0)(1)(1)(0)(1) &=& \omega^2+\omega \\ (0)(1)(2) &=& \omega^\omega \\ (0)(1)(2)(3) &=& \omega^{\omega^\omega} \\ (0)(1)(2)(3,0)(4,0) &=& \omega^{\omega^{\omega^\omega}} \\ \end{array}, \begin{array}{ll} (0)(1,1) &=& \varepsilon_0 \\ \end{array} (0,0)(1,1)(1,0)[4] = (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)[4] \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 2 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 3 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 4 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 5 \\ \end{array} (0,0)(1,1)(1,0) = \varepsilon_0 \omega = \omega^{\varepsilon_0+1} \begin{array}{ll} (0,0)(1,1)(1,0) &=& \varepsilon_0 \omega = \omega^{\varepsilon_0+1} \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \omega2 \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \omega3 \\ \end{array}, \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(1,0)(2,1) &=& \varepsilon_0^2 \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1) &=& \varepsilon_0^{\varepsilon_0} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1) &=& \varepsilon_0^{\varepsilon_0^2} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1) &=& \varepsilon_0^{\varepsilon_0^{\varepsilon_0}} \\ \end{array} (0,0)(1,1)(1,1) = \varepsilon_1 \begin{eqnarray*} (0,0)(1,1)(2,0) &=& \varepsilon_{\omega} \\ (0,0)(1,1)(2,0)(2,0) &=& \varepsilon_{\omega^2} \\ (0,0)(1,1)(2,0)(3,0) &=& \varepsilon_{\omega^\omega}) \\ (0,0)(1,1)(2,0)(3,1) &=& \varepsilon_{\varepsilon_0} \\ (0,0)(1,1)(2,0)(3,1)(4,0)(5,1) &=& \varepsilon_{\varepsilon_{\varepsilon_0}} \\ (0,0)(1,1)(2,1) &=& \zeta_0 \\ (0,0)(1,1)(2,1)(1,1) &=& \varepsilon_{\zeta_0+1} \\ (0,0)(1,1)(2,1)(1,1)(2,1) &=& \zeta_1 \\ (0,0)(1,1)(2,1)(2,0) &=& \zeta_\omega \\ (0,0)(1,1)(2,1)(2,1) &=& \eta_0 \\ (0,0)(1,1)(2,1)(2,1)(2,1) &=& \varphi(4,0) \\ (0,0)(1,1)(2,1)(3,0) &=& \varphi(\omega,0) \\ (0,0)(1,1)(2,1)(3,1) &=& \Gamma_0 &=& \psi_0(\Omega^\Omega) &=& \varphi(1,0,0) \end{eqnarray*} \begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(1,1) &=& \varepsilon_{\Gamma_0+1} \\ (0,0)(1,1)(2,1)(3,1)(1,1)(2,1) &=& \zeta_{\Gamma_0+1} \\ (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1) &=& \Gamma_1 &=& \varphi(1,0,1) \\ (0,0)(1,1)(2,1)(3,1)(2,0) &=& \Gamma_\omega &=& \varphi(1,0,\omega) \\ (0,0)(1,1)(2,1)(3,1)(2,1) &=& \varphi(1,1,0) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1) &=& \varphi(1,1,1) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(2,0) &=& \varphi(1,1,\omega) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(2,1) &=& \varphi(1,2,0) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(3,1) &=& \varphi(2,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,0) &=& \varphi(\omega,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1) &=& \varphi(1,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)(3,1) &=& \varphi(1,0,0,1) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1) &=& \varphi(1,0,1,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1) &=& \varphi(1,1,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(2,1)(3,1) &=& \varphi(1,2,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1) &=& \varphi(2,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1) &=& \varphi(3,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(3,0) &=& \varphi(\omega,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(3,1) &=& \varphi(1,0,0,0,0) &=& \psi_0(\Omega^{\Omega^3}) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(3,1)(3,1) &=& \varphi(1,0,0,0,0,0) &=& \psi_0(\Omega^{\Omega^4}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)&=& \psi_0(\Omega^{\Omega^\omega}) &&\text{(SVO)} \\ (0,0)(1,1)(2,1)(3,1)(4,0)(3,1) &=& \psi_0(\Omega^{\Omega^{\omega+1}}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)(4,0) &=& \psi_0(\Omega^{\Omega^{\omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)(5,0) &=& \psi_0(\Omega^{\Omega^{\omega^\omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)(5,1) &=& \psi_0(\Omega^{\Omega^{\varepsilon_0}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1) &=& \psi_0(\Omega^{\Omega^\Omega}) &&\text{(LVO)} \end{eqnarray*} \begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(4,1)(4,0) &=& \psi_0(\Omega^{\Omega^{\Omega \omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,1) &=& \psi_0(\Omega^{\Omega^{\Omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,0) &=& \psi_0(\Omega^{\Omega^{\Omega^\omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1) &=& \psi_0(\Omega^{\Omega^{\Omega^\Omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}}) \\ (0,0)(1,1)(2,2) &=& \psi_0(\varepsilon_{\Omega+1}) &=& \psi_1(0)) \end{eqnarray*} Etc. (0,0,0)(1,1,1) = (0,0)(1,1)...(n,n) = ψ_0(ε_{Ω_{ω}+1}) or ψ_0(Ω_ω) Use the Updated tier 9 -illions: Monogonillion = H1#9 Digonillion = H2#9 Trigonillion = H3#9 Tetragonillion = H4#9 Pentagonillion / Quintagonillion = H5#9 Hexagonillion / Sextagonillion = H6#9 Heptagonillion / Septagonillion = H7#9 Octagonillion = H8#9 Enneagonillion / Nonagonillion / Ennegonillion = H9#9 Decagonillion / Dekogonillion = H10#9 Unoedecagonillion = H11#9 Duoedecagonillion = H12#9 Tridecagonillion = H13#9 Quattadecagonillion = H14#9 Quinquadecagonillion = H15#9 Decasexgonillion = H16#9 Deca’eptegonillion = H17#9 Decaoctoegon = H18#9 Decaennegon = H19#9 Icosegonillion = H20#9 Icosoeunoegonillion = H21#9 … Triantegonillion = H30#9 Tetrantegonillion = H40#9 Penantegonillion = H50#9 Hextenatagonillion = H60#9 Heptenatagonillion = H70#9 Ogodenatagonillion = H80#9 Nonoeatagonillion = H90#9 Ekatoegonillion = H100#9 Doecosaegonillion = H200#9 Tricosaegonillion = H300#9 Quadracosaegonillion = H400#9 Pentacosaegonillion = H500#9 Hexacosaegonillion = H600#9 Ebdomoecosaegonillion = H700#9 Ogdoecosaegonillion = H800#9 Enneaecodaegonillion = H900#9 … Circleillion / Chillaegonillion = H1000#9 = H1#10 More BMS Examples: (0)(1,1,1)(2,1,1)(3,1) = (0,0,0)(1,1,1)(2,1,1)(3,1,0) = ψ_0(Ω_Ω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1) = ψ_0(Ω_Ω_Ω) … (0)(1,1,1)(2,1,1)(3,1)(2) = (0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0) = (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1,1)...(1,1,1)(2,1,1)(3,1,1) = Countable limit of Extended Buchholz’s function = Extended Buchholz’s Ordinal (0,0,0)(1,1,1)(2,2,2) = (0)(1,1,1)(2,2,2) = ψ(C(1{Sω}0)) in Trange Ink’s ExUNOCF (in War arena) > ψ_0(I) (0,0,0,0)(1,1,1,1) = (0)(1,1,1,1) = (0)(1,1,1,1) = ψ(C(1{Ω_c}0)) >> ψ_0(Μ) (0,0,0,0,0)(1,1,1,1,1) = (0)(1,1,1,1,1) = ψ_{Z}(ε_2) >> ψ_0(Ν) < ψ_0(T) (0,0,0,0,0,0)(1,1,1,1,1,1) = (0)(1,1,1,1,1,1) = ψ_{Z}(ε_3) > ψ_0(Τ) … (0,0,0,0,0,...)(1,1,1,1,1,...)... = Church Kleene Ordinal Inspired, with ordinals increasing and Remaked by the Full Ordinal LNGI correct JSON code: "final": "Γ0", "bits": [ [ "", "" ], [ "ω+", "" ], [ "ω2+", "" ], [ "ω3+", "" ], [ "ω", "" ], [ "ε0+", "" ], [ "ε0*(", ")" ], [ "ε0", "" ], [ "ε1*(", ")" ], [ "ε1", "" ], [ "ε2", "" ], [ "ε", "" ], [ "ζ0+", "" ], [ "ζ0*(", ")" ], [ "ζ0", "" ], [ "ε", "" ], [ "ζ1", "" ], [ "ε", "" ], [ "ζ", "" ], [ "η", "" ], [ "φ5(", ")" ], [ "φ6(", ")" ], [ "φ", "(0)" ] ], "limits": [ 0, 2000, 5000, 10000, 15000, 30000, 40000, 50000, 70000, 80000, 100000, 120000, 175000, 185000, 195000, 210000, 240000, 250000, 270000, 320000, 350000, 360000, 365000, 500000 ], "offsets": [ 0, 0, 0, 0, 1600, 0, 2000, 2000, 1333.3334, 2000, 1333.3334, 1500, 0, 2000, 2000, 175054.0541, 1000, 240013.3869, 1500, 0, 0, 0, 1750 ] } and // i made it BHO var num = 0; var speed = 1; var om = "<i>ω</i>"; var ep = "<i>ε</i>"; var ze = "<i>ζ</i>"; var ph = "<i>φ</i>"; var pz = "<i>ψ</i>0"; function dv(x){return 1/(1-x*0.99)} function ad(x){return x<1?"":"+"+x} function tw(x,y,t){return x<2?t:(t+["."+x,""+x+""][y])} function c1(x,y){ if (y<1) { if (x<1) { return ""+(dv(x)-1).toFixed(3)+"" } else return tw(Math.floor(dv(x%1)),0,tw(Math.floor(x),1,om))+(x<100?"+"+c1(dv(x%1)%1*Math.floor(x),0):"") } else if(x<1){ return c1(dv(x)-1,y-1) } else return tw(Math.floor(dv(x%1)),0,om+""+tw(y,0,om)+ad(Math.floor(x-1))+"")+(x<100?"+"+c1(dv(x%1)%1*Math.floor(x),y):"") } function oar(x,y){ return c1(0.51+(x*y)-0.51*x,0) } function c2(x,y){ return (om+"").repeat(y)+c1(x,0)+"".repeat(y) } function oth(x,p,y){ if (y<1){ return tw(Math.floor(x),0,om+""+om+""+p+"")+"+"+(x%1<0.5?c2(dv(x%1*2)-1,0):c2(1+(x%1-0.5)*2*(p-1),1)) } else return om+""+tw(Math.floor(x),0,tw(p,1,om))+"+"+(x<2?oar(x%1,p):c1(x%1*p,0))+"" } function c3(x,y,i){ return tw(Math.floor(x),0,(om+"").repeat(y)+om+"".repeat(y))+"+"+c2(dv((x*y)%1)-((x%1)*y<1?(x<2?(i<1?1:0.49):1):0),Math.floor((x%1)*y)) } function c4(x,i){ return x<1?c1(dv(dv(x)%1)+(dv(x)<2?0.26:0)*i,Math.floor(dv(x))):oth(dv(x*2%1),Math.floor(x)+1,Math.floor(x%1*2)) } function c3e(x,y,i){ if (y<2) { return c4(x-1,i) } else return c3(x,y,i) } function c5(x){ if (x<14) { let dw = x%1*Math.floor(x) return (om+"").repeat(Math.floor(dw))+c3e(dv(dw%1),Math.floor(x)-Math.floor(dw),dw<1?0:1)+"".repeat(Math.floor(dw)) } else return ""+(x+1).toFixed(1)+""+om } function eps(x){ if (x<1) { let dw2 = dv(x)-1 return dw2<14?c2(dv(dw2%1)-(dw2<1?1:0),Math.floor(dw2)):(""+(dw2+1).toFixed(1)+""+om) } else return tw(Math.floor(dv(x%1)),0,x<2?(ep+"0"):(om+""+ep+"0+"+Math.floor(x-1)+""))+"+"+eps((dv(x%1)%1)*Math.floor(x)) } function epsd(x){ if (x<1) { return eps(dv(x)) } else if (x<14) { return (om+"").repeat(Math.floor(x))+eps(dv(x%1)+0.25571)+"".repeat(Math.floor(x)) } else return ep+"1 ["+x.toFixed(1)+"]" } function c6(x,y,p){ if (p<8) { if (y<1) { if (x<1) { let gh = dv(x)-1; return p<1?(gh<10?c2(dv(gh%1)-(gh<1?1:0),Math.floor(gh)):(""+(gh+1).toFixed(1)+""+om)):(c6(dv(gh%1)-1+(gh<1?0:[1.25568,1.17231,1.13058,1.10554,1.08883,1.076902,1.067958][p-1]),Math.floor(gh),p-1)) } else return tw(Math.floor(x),0,ep+""+p+"")+"+"+c6(x%1,0,p) } else return y<10?(om+""+c6(x,y-1,p)+""):(ep+""+(p+1)+"["+y+"]") } else return ep+""+p+"" } function esl(x) { return x<7?c6(0.33897+(x%1)*0.66103,0,Math.floor(x)+1):ep+""+x.toFixed(2)+""; } function c7(x) { if (x<1) { return dv(x)<15?c2(dv(dv(x)%1)-(dv(x)<2?1:0),Math.floor(dv(x)-1)):(""+dv(x).toFixed(1)+""+om) } else return esl(x-1) } function eps2(x){ if (x<1) { let dw2 = dv(x)-1 return c7(dw2) } else return tw(Math.floor(dv(x%1)),0,x<2?(ep+""+om+""):(om+""+ep+""+om+"+"+Math.floor(x-1)+""))+"+"+eps2((dv(x%1)%1)*Math.floor(x)) } function epsd2(x){ if (x<1) { return eps2(dv(x)) } else if (x<14) { return (om+"").repeat(Math.floor(x))+eps2(dv(x%1)+0.2057)+"".repeat(Math.floor(x)) } else return ep+""+om+"+1 ["+x.toFixed(1)+"]" } function everyc7(x){ return x<1?c1(dv(x)-1,0):x<2?c5(dv(x-1)):x<3?epsd(dv(x-2)-1):x<4?esl(dv(x-3)):epsd2(dv(x-4)-1) } function getzeta(x){ if (x<1) { return everyc7(x*5) } else { let dw3 = Math.floor(x) return x<10?((ep+"").repeat(dw3)+everyc7((x%1)*(5-0.57824)+0.57824)+"".repeat(dw3)):(ze+"0 ["+x.toFixed(2)+"]") } } function pht(y,x,d){ if (y<4){ return ("<i>"+"εζη"[y-1]+"</i>"+"").repeat(d)+x+"".repeat(d) } else return (ph+""+y+" (").repeat(d)+x+")".repeat(d) } function phi(x){ if (x<1) { let fh = dv(x)-1; return fh<9?c2(dv(fh%1)-(fh<1?1:0),Math.floor(fh)):(""+(fh+1).toFixed(1)+""+om); } else { let fh = dv(x%1); return fh<10?(pht(Math.floor(x),phi(fh%1*Math.floor(x)),Math.floor(fh))):(pht(Math.floor(x+1),0,1)+"["+fh.toFixed(1)+"]") } } function phia(x,y){ if (x<1) { let fh = dv(x); return tw(Math.floor(fh),0,pht(y,0,1))+"+"+phi(y<2?(fh%1):(fh%1<0.5?(fh%1*2):(1+(fh%1-0.5)*((y-1)*2)))) } else { let fh = dv(x%1); let x2 = Math.floor(x); return fh<10?(x<2?((om+"").repeat(Math.floor(fh))+phia((y<2?0.20568:0.1145)+(fh%1)-((y<2?0.20568:0.1145)*(fh%1)),y)+"".repeat(Math.floor(fh))):pht(x2-1,phia(0.1145+(fh%1*x2)-(0.1145*(fh%1)),y),Math.floor(fh))):((x<y?pht(x2,pht(y,0,1)+"+1",1):pht(y,1,1))+"["+fh.toFixed(1)+"]"); } } function phib(x,y,c){ if (x<1) { let fh = dv(x); return tw(Math.floor(fh),0,pht(y,c,1))+"+"+(fh%1<0.75?phi(y<2?(fh%1):(fh%1<0.5?(fh%1*2):(1+((fh%1-0.5)*2)*((y-1)*2)))):pht(y,((fh%1-0.75)*4*c).toFixed(3),1)) } else { let fh = dv(x%1); let x2 = Math.floor(x); return fh<10?(x<2?((om+"").repeat(Math.floor(fh))+phib((y<2?0.20568:0.1145)+(fh%1)-((y<2?0.20568:0.1145)*(fh%1)),y,c)+"".repeat(Math.floor(fh))):pht(x2-1,phib(0.1145+(fh%1*x2)-(0.1145*(fh%1)),y,c),Math.floor(fh))):((x<y?pht(x2,pht(y,c,1)+"+1",1):pht(y,1+c,1))+"["+fh.toFixed(1)+"]"); } } function phiAl(x){ let fh = dv(x%1)-1; if (x<1) { return fh<9?c2(dv(fh%1),Math.floor(fh)):(""+(fh+1).toFixed(3)+""+om) } else { let x2 = Math.floor(x); if (fh<1) { return phia(fh%1*(x2+1),x2) } else return phib(fh%1*(x2+1),x2,Math.floor(fh)) } } function phiCr(x){ if (x<1) { return phiAl(x) } else if (x%1<0.5) { return phiAl(Math.floor(x)+(x*2)%1) } else { let fh = dv(((x%1)-0.5)*2) return fh<10?pht(Math.floor(x),phiCr((fh%1)*(Math.floor(x)+0.5)),Math.floor(fh)):(pht(Math.floor(x+1),0,1)+"["+fh.toFixed(2)+"]") } } function phiOm(x,y){ if (y<1){ return phi(x) } else if (x<1){ return phiOm(dv(x)-1,y-1) } else { let fh = dv(x%1); let fh2 = Math.floor(fh); if (fh<10) { return (ph+""+tw(y,0,om)+ad(Math.floor(x)-1)+" (").repeat(fh2)+phiOm((fh%1)*Math.floor(x),y)+(")").repeat(fh2) } else return ph+""+tw(y,0,om)+ad(Math.floor(x))+" (0)["+fh.toFixed(2)+"]" } } function phiAp(x){ if (x<1) { return x<0.1?c1(x*10,0):phiCr(dv((x-0.1)*(1/0.9))-1) } else if (x<2) { return phiOm(dv(dv(x-1)%1),Math.floor(dv(x-1))) } else if (x<3) { return ph+""+c1(dv(x-2)+1,0)+" (0)" } else if (x<4) { let fh = dv(x-3); let fh2 = Math.floor(fh); if (fh<10) { return (ph+""+om+""+om+" (").repeat(fh2)+phiAp(fh%1*3)+")".repeat(fh2) } else return ph+""+om+""+om+"+1 (0)["+fh.toFixed(2)+"]" } else if (x<5) { return ph+""+c5(dv(x-4)+0.14847)+" (0)" } else if (x<6) { let fh = dv(x-5); let fh2 = Math.floor(fh); if (fh<10) { return (ph+""+ep+"0 (").repeat(fh2)+phiAp(fh%1*5)+")".repeat(fh2) } else return ph+""+ep+"0+1 (0)["+fh.toFixed(2)+"]" } else if (x<7) { return ph+""+phiCr(dv(x-6)+0.0471)+" (0)" } } function phiAC(x){ if (x<9) { return (ph+"").repeat(Math.floor(x))+phiAp((x%1)*6+1)+" (0)".repeat(Math.floor(x)) } else return "Γ0 ["+(x+1).toFixed(2)+"]" } function PH(x){ if (x<10) { return x<1?phi(dv(x)-1):((ph+"").repeat(Math.floor(x))+phi(dv(x%1)-1+0.33897)+" (0)".repeat(Math.floor(x))) } else return "Γ0["+x.toFixed(1)+"]" } function Gam(x,y) { if (y<1) { if (x<1) { return PH(dv(x)-1) } else if (x<2) { let fh = dv(x-1); return tw(Math.floor(fh),0,"Γ0")+"+"+Gam(fh%1,0) } else if (x<3) { let fh = dv(x-2); return fh<10?((om+"").repeat(Math.floor(fh))+Gam(1.14847+(fh%1)-0.14847*(fh%1),0)+"".repeat(Math.floor(fh))):(pht(1,"Γ0+1",1)+"["+fh.toFixed(1)+"]") } else { let fh = dv(x%1); return fh<10?pht(Math.floor(x-2),Gam(1.14847+((fh%1)*Math.floor(x-1))-0.14847*(fh%1),0),Math.floor(fh)):(pht(Math.floor(x-1),"Γ0+1",1)+"["+fh.toFixed(1)+"]") } } else if (x<1) { let fh = dv(x)-1; return Gam(fh,0) } else if (x<2) { let fh = dv(x-1)-1; return ph+""+PH(fh+0.25571)+" (Γ0+1)" } else if (x<3) { let fh = dv(x-2); return fh<10?((ph+"Γ0 (").repeat(Math.floor(fh))+Gam(0.14847+(fh%1*2)-0.14847*(fh%1))+")".repeat(Math.floor(fh))):(ph+"Γ0+1 (0)["+fh.toFixed(1)+"]") } else if (x<4) { let fh = dv(x-3); return fh<10?((ph+"").repeat(Math.floor(fh))+Gam(0.539953+(fh%1*3)-0.539953*(fh%1))+" (0)".repeat(Math.floor(fh))):("Γ1["+fh.toFixed(1)+"]") } } function PhD(x,y,g) { let x2 = Math.floor(x); if (x<1) { if (y<2) { return PH(dv(x)-1+0.25571*g) } else return x<0.5?PhD(dv(x*2)-1,y-1,g):dv(x*2-1)<10?((ph+"("+(y-1)+", ").repeat(Math.floor(dv(x*2-1)))+PhD(dv(dv(x*2-1)%1)-1,y-1,1)+", 0)".repeat(Math.floor(dv(x*2-1)))):(ph+"("+y+", 0, 0)["+dv(x*2-1).toFixed(1)+"]") } else if (x<3 && y<2) { let fh = dv(x%1); return fh<10?(("Γβα"[Math.floor(x-1)]+"").repeat(Math.floor(fh))+PhD(fh%1*x2,y,0)+"".repeat(Math.floor(fh))):("Γβα"[Math.floor(x)]+"0["+fh.toFixed(1)+"]") } else { let fh = dv(x%1); return fh<10?((ph+"("+y+", "+Math.floor(x-1)+", ").repeat(Math.floor(fh))+PhD(fh%1*x2,y,0)+")".repeat(Math.floor(fh))):(ph+"("+y+", "+Math.floor(x)+", 0)["+fh.toFixed(1)+"]") } } function AStack(x){ let fh = dv(x%1); if (x<10){ return (ph+"(").repeat(Math.floor(x))+PhD(dv(fh%1)-1+(fh<2?0.2057:0.5),Math.floor(fh),0)+", 0, 0)".repeat(Math.floor(x)) } else return "AO["+x.toFixed(1)+"]" } function phiP(x,g){ let td = [0,0.2557,0.29192][g] if (x<1) { return phi(x+td-(td*x)) } else { let fh = dv(x%1); return fh<10?((ph+"(").repeat(Math.floor(fh))+phiP(fh%1*Math.floor(x),1)+(", 0".repeat(Math.floor(x))+")").repeat(Math.floor(fh))):(ph+"(1"+", 0".repeat(Math.floor(x)+1)+")["+fh.toFixed(1)+"]") } } function phiQ(x,y,g){ if (x<1) { return phiP(x*(y+1),g) } else { let fh = dv(x%1); if (fh<10) { if (x-y<2) { return (ph+"(1, "+"0, ".repeat(y-(Math.floor(x-1)))).repeat(Math.floor(fh))+phiQ(fh%1*Math.floor(x),y,(x<2?0:1))+(", 0".repeat(Math.floor(x-1))+")").repeat(Math.floor(fh)) } else return (ph+"(").repeat(Math.floor(fh))+phiQ(fh%1*Math.floor(x),y,2)+(", 0".repeat(y+1)+")").repeat(Math.floor(fh)) } else if (x-y<2) { if (x-y<1) { return ph+"(1, "+"0, ".repeat(y-Math.floor(x))+"1"+", 0".repeat(Math.floor(x))+")["+fh.toFixed(1)+"]" } else return ph+"(2"+", 0".repeat(y+1)+")["+fh.toFixed(1)+"]" } else return ph+"(1"+", 0".repeat(y+2)+")["+fh.toFixed(1)+"]" } } function ReachSVO(x){ if (x<18) { return phiQ(1+(x%1)*Math.floor(x+2),Math.floor(x),0) } else return "SVO["+(x+2).toFixed(2)+"]" } function Level(x,y,g) { let x2 = Math.floor(x); if (y==0) { return phi(x+0.2557) } else if (y==1) { return x<20?phiP(x+1.10165,0):("SVO["+x.toFixed(2)+"]") } else if (y==2) { if (g<1) { let fh = x%1*2 return x<10?((pz+"(ΩΩ"+om+"").repeat(x2)+Level(dv(fh%1)-1,Math.floor(fh),0)+")".repeat(x2)):(pz+"(ΩΩ"+om+"+1)["+x.toFixed(2)+"]") } else return x<10?((pz+"(ΩΩ"+om+"+"+g+"").repeat(x2)+Level(dv(x%1)-1,2,g-1)+")".repeat(x2)):(pz+"(ΩΩ"+om+"+"+(g+1)+")["+x.toFixed(2)+"]") } else if (y==3) { let fh = dv(x%1)-1; if (x2<10){ return (pz+"(ΩΩ").repeat(x2)+Level(dv(fh%1)-(fh<1?0.943568:0),2,Math.floor(fh))+")".repeat(x2) } else return "LVO["+x.toFixed(1)+"]" } else if (y==4) { if (x<0.5) { return (dv(x*2)<2?"":tw(Math.floor(dv(x*2))-1,0,om)+"+")+(dv(dv(x*2)%1)-1).toFixed(3) } else return Level(0.00975+dv(x*2-1)-1,3,0) } } function leti(x,p,r,r2,r3){ if (x<1){ return Level(x,4,0) } else { let x1 = dv(x%1); if (x1<10) { return p[Math.floor(x-1)].repeat(Math.floor(x1))+leti((x1%1)*Math.floor(x)+r3[Math.floor(x-1)]*(1-(x1%1)),p,r,r2,r3)+r[Math.floor(x-1)].repeat(Math.floor(x1)) } else return r2[Math.floor(x-1)]+"["+(x1).toFixed(1)+"]" } } function comth(x) { return leti(x, [pz+"(ΩΩΩ", pz+"(ΩΩΩ+", pz+"(ΩΩΩ+Ω", pz+"(ΩΩΩ", pz+"(ΩΩΩ+1", pz+"(ΩΩΩ+2", pz+"(ΩΩΩ+", pz+"(ΩΩΩ2+", pz+"(ΩΩΩ.(", pz+"(ΩΩΩ2+", pz+"(ΩΩΩ2+Ω.(", pz+"(ΩΩΩ2.(", pz+"(ΩΩΩ"], [")",")",")",")",")",")",")",")", "))",")","))","))",")"], [pz+"(ΩΩΩ+1)", pz+"(ΩΩΩ+Ω)", pz+"(ΩΩΩ2)", pz+"(ΩΩΩ+1)", pz+"(ΩΩΩ+2)", pz+"(ΩΩΩ+3)", pz+"(ΩΩΩ2)", pz+"(ΩΩΩ3)", pz+"(ΩΩΩ2)", pz+"(ΩΩΩ2+Ω)", pz+"(ΩΩΩ2.2)", pz+"(ΩΩΩ3)", pz+"(ΩΩΩΩ)"], [0.16946,0.16946,0.16946,0.20323,0.16946,0.16946,0.21769,0,0.21769,0,0.16946,0.20323,0.21769]) } function reachBHO(x){ let x2 = Math.floor(x) if (x<1) { let zk = dv(x)-1; return zk<11?(c2(dv(zk%1)-(zk<1?1:0),Math.floor(zk))):(""+zk.toFixed(1)+""+om) } else { let ch1 = Math.floor((x-1)%4) let ch2 = Math.floor((x-1)/4) let zk = dv(x%1); if (ch1==0) { return zk<10?((pz+"("+"Ω".repeat(ch2)+"Ω+").repeat(Math.floor(zk))+reachBHO((zk%1)*x2)+("".repeat(ch2)+")").repeat(Math.floor(zk))):(pz+"("+"Ω".repeat(ch2)+"Ω.2"+"".repeat(ch2)+")["+(zk).toFixed(1)+"]") } if (ch1==1) { return zk<10?((pz+"("+"Ω".repeat(ch2)+"Ω.(").repeat(Math.floor(zk))+reachBHO((zk%1)*x2+(0.29192-(zk%1)))+(")"+"".repeat(ch2)+")").repeat(Math.floor(zk))):(pz+"("+"Ω".repeat(ch2)+"Ω2"+"".repeat(ch2)+")["+(zk).toFixed(1)+"]") } if (ch1==2) { return zk<10?((pz+"("+"Ω".repeat(ch2)+"Ω2.(").repeat(Math.floor(zk))+reachBHO((zk%1)*x2+(0.2557-(zk%1)))+(")"+"".repeat(ch2)+")").repeat(Math.floor(zk))):(pz+"("+"Ω".repeat(ch2)+"Ω3"+"".repeat(ch2)+")["+(zk).toFixed(1)+"]") } if (ch1==3) { return zk<10?((pz+"("+"Ω".repeat(ch2)+"Ω").repeat(Math.floor(zk))+reachBHO((zk%1)*x2+(0.30639-(zk%1)))+(""+"".repeat(ch2)+")").repeat(Math.floor(zk))):(pz+"("+"Ω".repeat(ch2)+"Ω2"+"".repeat(ch2)+")["+(zk).toFixed(1)+"]") } } } function current(x){ if (x<100000){ let i = x/100000 return c1(dv(i)-1,0) } else if (x<500000){ let i = (x-100000)/400000 return c5(dv(i)) } else if (x<550000) { let i = (x-500000)/50000 return eps(dv(i)) } else if (x<850000) { let i = (x-550000)/300000 return epsd(dv(i)) } else if (x<1250000) { let i = (x-850000)/400000 return esl(dv(i)) } else if (x<1260000) { let i = (x-1250000)/10000 return epsd2(dv(i)-1) } else if (x<1800000) { let i = (x-1260000)/540000 return getzeta(dv(i)) } else if (x<2500000) { let i = (x-1800000)/700000 return phiCr(1+dv(i)) } else if (x<3100000) { let i = (x-2500000)/600000 return phiAp(i*6+1) } else if (x<4000000) { let i = (x-3100000)/900000 return phiAC(dv(i)) } else if (x<4200000) { let i = (x-4000000)/200000 return Gam(0.5051+(i*4)-0.5051*i,1) } else if (x<4800000) { let i = (x-4200000)/600000 return AStack(dv(i)-1+0.33226) } else if (x<6000000) { let i = (x-4800000)/1200000 return ReachSVO(1+dv(i)) } else if (x<7000000) { let i = (x-6000000)/1000000 return comth(i*14+0.623191-(0.623191*i)) } else if (x<7200000) { let i = (x-7000000)/200000 return reachBHO(12+dv(i)) } else return "BHO = "+pz+"(Ω2)" } setInterval(_=>{ document.getElementById("number").innerHTML = current(num).replace("+0.000",""); document.getElementById("speed").innerHTML = "Speed: "+speed num += speed; }) Use AAS (Aarex’s Abbreviation System) for Abbreviations: “Like Standard, but the parts are changed. Also in standard, there is a space after the number then you have the abbreviation for it (ex: 10 Qa, 123.45 Sp). However in AAS, the space is after the number (ex: 10 Qa, 123.45 Sp). Unit parts: K, M, B, T, Qa, Qi, Sx, Sp, Oc, and No. Units place: U, D, T, Qt, Qn, Sx, Sp, O, N Tens place: D/De, V/Vg, Tg, Qd, Qq, Sc, St, Oe, Ng Hundreds place: C/Ce, Dc, Tc, Qe, Qu, Se, Si, Oe, Ne Special roots: Mi, Mc, Na, Pc, Fem, At, Zep, Yo, etc. D and De are toggled using the notation options. Killillion = 10^(3x10^3000+3) … Killo-decillion = 10^(3x10^3000+33) etc. Tria-teraksys = 10^^^3 Tetra-teraksys = 10^^^4 … Deka-teraksys = 10^^^10 … Gaggol / Hecta-teraksys = 10^^^100 … Tria-petaksys = 10^^^^3 or 10{4}3 Grahal = 3^^^^3 … Deka-petaksys = 10^^^^10 or 10{4}10 … Geegol = 10{4}100 … Deka-exaksys = 10{5}10 Deka-eptaksys = 10{6}10 Deka-octaksys = 10{7}10 Deka-ennaksys / Deka-enneaksys = 10{8}10 … Deka-endekaksys = 10{10}10 = Tridecal Boogol = 10{100}10 … General = {10,10,10,10} = 10{{{{{{{{{{10}}}}}}}}}}10 = 10{10}^(10) 10. Graham grahal = {3,3,{3,3,4}} = 3{3{4}3}3 … Forcal = f_{\omega+1}(1000000) … G_G_… (Graham’s number times) = f_{Φ}(1) = f_{ω+2}(Graham’s number) f_{Φ}(2) … f_{Φ_{Φ}}(1) f_{Φ_{Φ…}}(1) (187196 times) = TREE(3). TREE(n) = f_{\theta(\Omega^{\omega}\omega)}(n) Examples using a theta ordinal notation called “Transfinitary theta function”: θ(Ω^ω) = SVO θ(Ω^Ω) = LVO θ(ε_{Ω+1}) = ΒΗΟ .. θ(Γ_{Ω+1}) = ψ_0(Ω_2^{Ω_2}) = ψ(Γ_{Ω+1}) … θ(Θ(1)) = θ(Ω_Ω_…) = θ(Λ_0) and so on… the transfinitary theta function goes on forever… \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) = Small Rathjen Ordinal