User blog:Nayuta Ito/Is this bigger than Rayo's number?

This number is from a Japanese googology competition held (technically) a month ago. I am just translating other person's number, so my translation might be wrong. According to the original author, it's bigger than BIG FOOT.

Number:

The alphabets of a formal language FOL is defined as followings:

$$\mathrm{FOL_{constant}}:=\{2n+1|n\in\mathbb{N}\}$$

$$\mathrm{FOL_{variable}}:=\{2(2n+1)|n\in\mathbb{N}\}$$

For a natural number $$a$$, $$\mathrm{FOL_{a,\ argument\ function}}:=\{4(2^{a+1}(2n+1)+1)|n\in\mathbb{N}\}$$

For a natural number $$a$$, $$\mathrm{FOL_{a,\ argument\ relation}}:=\{8(2^{a+1}(2n+1)+1)|n\in\mathbb{N}\}$$

$$\mathrm{FOL_{logical}}:=\{16,48,80\}$$

$$\mathrm{FOL_{contradiction}}:=\{32\}$$

$$\mathrm{FOL_{logical\ formula}}:=\{64(2n+1)|n\in\mathbb{N}\}$$

And the syntax is as follows:


 * 1) A term in FOL is a string in FOL which is an element of $$\mathrm{FOL_{constant}}$$; an element of $$\mathrm{FOL_{variable}}$$; or $$f(t_1,t_2,\cdots,t_a)$$, where $$a$$ is a natural number, f is an element of $$\mathrm{FOL_{a,\ argument\ function}}$$, and $$t_1,t_2,\cdots t_n$$ are terms.
 * 2) A logical formula in FOL is a string in FOL which is an element of $$\mathrm{FOL_{logical\ formula}}$$; 32; 16x(P) where x is an element of $$\mathrm{FOL_{variable}}$$; (P)48(Q) where P and Q are formulae; 80(P) where P is a formula; or $$f(t_1,t_2,\cdots,t_a)$$, where $$a$$ is a natural number, f is an element of $$\mathrm{FOL_{a,\ argument\ relation}}$$, and $$t_1,t_2,\cdots t_n$$ are terms.
 * 3) A logical formula in FOL is first-order if and only if it does not contain an element of $$\mathrm{FOL_{logical\ formula}}$$ in its string.

Especially, the formal language FOL', FOL without $$\mathrm{FOL_{logical\ formula}}$$, is a formal language of first-order predicate logic, and the following is a logical formula in FOL: ZFC axioms combined by AND operator in FOL.

For a logical formula A and first-order logical formula P in FOL, P is provable from the axiom A means P is provable in FOL' using (finite first-order logical formula obtained by replacing all elements of $$\mathrm{FOL_{logical\ formula}}$$ into some concrete first-order logical formulae in A) as an axiom.

For a logical formula A in FOL, A is consistent means that 32 is not provable from the axiom A.

For a natural number n, a Peano arithmetic term tn is defined as follows:
 * If n=0, tn is 0 (as a constant).
 * If n>0, tn is S(t n-1/sub>) (as a term).

For a natural number n, the natural number f(n) is defined as follows:

The smallest minimum number bigger than any natural number m such that:

There exists a logical formula A and first-order logical formula P in FOL, both of which have less than n symbols, such that:

A connotes Peano arithmetic, if A_ZFC is consistent, then A is inconsistent, there exists only one element of $$\mathrm{FOL_{variable}}$$ (call it x) such that it's not bounded by 16, and m is the smallest natural number where 80(Q) is not provable with A, where Q is:

P, whose all x without bounded by 16 are replaced by t. the image of tm in A, and 16x((P)48(x=t)) combined by AND operator in FOL.

"fight for not being able to write the smallest proof" number is defined as f(10^100).