Large Number Garden Number

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The Large Number Garden Number (Japanese: 巨大数庭園数, also abbreviated LNGN) is the abbreviated name of the large number equal to \(f^{10}(10 \uparrow^{10} 10)\), where \(f\) is the function defined in First Order Theory beyond Higher Order Set Theory. ^[1]^[2] The term was coined by Googology Wiki user P進大好きbot. The original name of LNGN in Japanese is translated into English as

Come on, friends, the large number garden is finally complete! Let me explain the function of this garden. The first is the determination function of the address and the floor plan. When a character string is read, it automatically determines which miniature garden address it represents and in which miniature garden the floor plan of a large number garden can be reproduced. The second is the floor plan analysis function. If you specify the address of the miniature garden and read the floor plan of the reproducible large number garden there, it will tell you the large number that the garden can produce. The third important function is the ability to generate large numbers. Once a natural number is entered, all character strings within the upper limit of the number of characters are searched, and each is read into the address and floor plan determination function, leaving only the reproducible floor plan for each miniature garden. By enumerating them and loading them into the analysis function of the floor plan, you can obtain the large numbers that they can produce, and by putting them all together, you can create new large numbers! Huh? Can you really get a large number with that? As usual, my ally is skeptical. But hey, here's the floor plan for the large number garden itself. If you load this into the analysis function, it will tell you how large numbers you can generate. Huh? How many characters does this floor plan have? What's the use of knowing such things?

This number can be considered to be the largest well-defined googologism which is not a salad number.

Definition[]

Theory[]

First, a language L is defined by adding a 1-ary function symbol U to the language of first order set theory with countably many variable term symbols and the set-membership relation symbol ∈. Define ZF_L as the set of L-formulae belonging to the axiom of ZF set theory. Here, the axiom schema of comprehension and replacement in ZF_L are parametrized by all L-formulae, i.e. formulae which can include U. Define a formal language L of first order logic by adding countably many constant term symbols, countably many function symbols, countably many relation symbols, and a new 1-ary function symbol Θ to an explicit formalization of L using an explicit Gödel correspondence. Then, we denote by ZFC_L the set of L-formulae belonging to the axiom of ZFC set theory, Here, the axiom schema of comprehension and replacement in ZFC_L are parametrized by all L-formulae, i.e. formulae which can include the formalization of U, additional constant term symbols, additional function symbols, additional relation symbols, and Θ. We explicitly encode ordinals below ε₀ and L-formulae into natural numbers in ZFL, and formalize the Henkin axiom "If there exists an x satisfying P, then Θ(n) satisfies P", for each variable term symbol x, each L-formula P with code n formalized into ZFC_L by the repetition of successor operations,

Denote by ZFCH_L the theory ZFC_L augmented by the Henkin axiom schema. The new function symbol Θ plays a role of "a family of Henkin constants". Please do not confound the base theory ZF_L and the formalized theory ZFCH_L. Denote by U1 the L-formula "For any ordinal α, U(α)⊨ZFCH_L". Under ZF_L augmented by {U1}, U(α) forms a model of ZFC_L, and hence forms an L-structure for any ordinal α. We denote by U^U(α) the interpretation of U in U(α). We denote by U2 the L-formula "For any ordinal α and any β∈α, U^U(α)(β)=U(β)", and by U3 the L-formula "For any ordinal α, there exists an ordinal β such that |U(α)|=V_β and for any x∈V_β and any y∈V_β, x∈^U(α)y is equivalent to x∈y", where V_β denotes the von Neumann hierarchy. Define T as the set ZF_L∪{U1,U2,U3} of L-formulae.

Embedding[]

By assigning to each atomic formula x_i∈x_j in ZFC set theory the L-formula (x_i∈x_j)∧(x_j∈U(0)), the theory T can be regarded as an extension of ZFC set theory. In particular, the set N defined in ZFC set theory is interpreted at U(0) as a term of T, which coincides with the term N defined under ZF_L because U(0) is a transitive model of ZFC_L. Therefore a large number definable in ZFC set theory is also definable in T, and forms a term which is a large number. Moreover, since L admits countably infinitely many constant term symbols, function symbols, and relation symbols, even a closed formula in a theory given by adding countably many constant term symbols, function symbols, and relation symbols to ZFC set theory can be interpreted at U(0) as a closed formula in T. Further, by assigning to each atomic formula x_i∈x_j in unsorted MK set theory the L-formula (x_i∈x_j)∧(x_j∈U(0)), the theory T can be regarded as an extension of MK set theory. Roughly speaking, U(0) formally plays a role of the universe of the first order set theory, the power set of U(0) formally plays a role of the universe of the second order set theory and the first order class theory, and its power set formally plays a role of the universe of the third order set theory. Since all of them are included in U(1), U formally plays a role of a strictly increasing sequence of the universes of higher order set theories. Note that the existence of such a strictly increasing sequence can be constructed in ZFC set theory augmented by Grothendieck universe axiom, which appears in usual mathematics.

Large Number[]

Explicitly define a surjective map: CNF: N→ε₀; i↦CNF(i) using Cantor normal form. For an L-formula P, denote by IsDefinition(P) the L-formula "There exists an x such that P and for any i, (P)[i/x] implies i=x". Denote by Definable(m,i,P) the L-formula "i∈N, P is an L-formula, U(CNF(i))⊨IsDefinition(P), and U(CNF(i))⊨(P)[m/x]", where m in (P)[m/x] is regarded as a parameter in an explicit way. For an n∈N, define f(n) as the sum of m∈N satisfying i∈n, P∈n, and Definable(m,i,P), In this way, there is finally an uncomputable large function f: N→N; n↦f(n). From here, the Large Number Garden Number is \(f^{10}(10 \uparrow^{10} 10)\).

External links[]

koteitan, pdf-translation of the original source, Dropbox.
koteitan, 巨大数寄席 Part 2, ニコニコ動画. (A video reading 巨大数庭園数 aloud.)
西宮七南-解説channel, ゆっくりと学ぶ巨大数論～巨大数庭園数～, YouTube. (A video introducing 巨大数庭園数.)

Sources[]

↑ 高階集合論を超えた1階述語論理 by p進大好きbot (the Japanese first source of the definition)
↑ First Order Theory beyond Higher Order Set Theory by P進大好きbot (the English summary of the first source)

Googology in Japan

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea

Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system

Implementation of existing works into programs

By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory

Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud