Large Number Garden Number

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. 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) 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 ZFL as the set of L-formulae belonging to the axiom of ZF set theory. Here, the axiom schema of comprehension and replacement in ZFL 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 ZFCL the set of L-formulae belonging to the axiom of ZFC set theory, Here, the axiom schema of comprehension and replacement in ZFCL 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 ZFCL by the repetition of successor operations,
Denote by ZFCHL the theory ZFCL 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 ZFL and the formalized theory ZFCHL. Denote by U1 the L-formula "For any ordinal α, U(α)⊨ZFCHL". Under ZFL augmented by {U1}, U(α) forms a model of ZFCL, and hence forms an L-structure for any ordinal α. We denote by UU(α) the interpretation of U in U(α). We denote by U2 the L-formula "For any ordinal α and any β∈α, UU(α)(β)=U(β)", and by U3 the L-formula "For any ordinal α, there exists an ordinal β such that |U(α)|=Vβ and for any x∈Vβ and any y∈Vβ, x∈U(α)y is equivalent to x∈y", where Vβ denotes the von Neumann hierarchy. Define T as the set ZFL∪{U1,U2,U3} of L-formulae.
=== Embedding ===
By assigning to each atomic formula xi∈xj in ZFC set theory the L-formula (xi∈xj)∧(xj∈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 ZFL because U(0) is a transitive model of ZFCL. 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 xi∈xj in unsorted MK set theory the L-formula (xi∈xj)∧(xj∈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 ==
Category:Numbers
Category:Uncomputable
Category:Googology in Asia
ja:巨大数庭園数
Category:Numbers based off of Rayo's number