Large Number Garden Number

The Large Number Garden Number (Japanese: 巨大数庭園数) 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. The term was coined by Googology Wiki user P進大好きbot.

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

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. I denoted 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. Explicitly encode ordinals below ε0 and L-formulae into natural numbers in ZFL. 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. I denoted by UU(α) the interpretation of U at U(α). I denoted 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. I 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 \(f^{10}(10 \uparrow^{10} 10)\).