User blog comment:Nayuta Ito/Is this bigger than Rayo's number?/@comment-27513631-20180504234554/@comment-35470197-20180505014941

> Emlightened

I am the one who defined the number.

First of all, I doubt that the original definition of Rayo's number is not well-defined in the following reasons:

- In order to have the truth value of Sat([\phi(x_1)],s) in the definition of Rayo's function make sense, one needs to fix a model V.

- In order to fix V, one needs to declare axioms of the base meta theory A.

- In order to assign a meta natural number m in A to a variable x_1 in FOST, one needs to define a way to construct a correspondence between meta natural numbers and elements in V. However, since no axiom for the model V is declared, one does not have such a canonical way. (If one assumes ZF on V, then the notion of natural numbers can be defined in a canonical way.)

- In order to define Rayo's number, which is defined as a meta natural number in A, one needs to assume a very strong axiom of replacement in A or V, because the class of Rayo-namable V-set is not trivially corresponds to a V-set. The choice of such a consistent axiom is not unique, and hence one have to declare an explicit choice. (At least, is there a known one?)

Therefore, when one compares a well-defined number with Rayo's number, he or she needs to give an alternative definition. I also did so. What do you think about it? (I know that such problems are written in the talk page. But in order to argue in a suitable way, I would like to know how you interprete the definition of Rayo's number.)

> The definition is very inelegant

Exactly. I am sorry for it. I used this notation because my number is an entry of a certain contest, where a base meta theory is declared. So in order to clearly distinguish formulae in meta language from those in FOL, I used such specific convention.

> The system is proof-based and is hence much weaker than Rayo's number.

It depends on the choice of an alternative well-defined Rayo's number. How do you define it? I simply used tautology under the assumption of several axioms of set theory which is necessary to ignore the problems above. Then by the completeness theorem, Rayo's number is deduced to a number defined in such a proof-based method.

I emphasise that the provability of a formula in A is a meta proposition, and I do not assume the provability of the provalibity in the base meta theory. So it is different from numbers such as "the least natural number n such that it is provable that n is bigger than any natural number satisfying..."

> The system, as translated, isn't really well-defined - there are no definitions of what 16, 48 and 80 are? I assume that 16 is xor, 48 is uniqueness quantifier, and 80 is true, as these are the obvious choices.

I used one of a usual convention for first-order logic; there are a single 2-ary logical operator "\to", a single 1-ary logical operator "\neg", and a single quantifier "\exists". Of course, you can choose other operators, because it makes little change of the number.

> A is used both as an arbitrary axiom and as Peano arithmetic, so it's unclear what this actually means.

No.

> A_ZFC is not the standard notation of any theory, regardless of A.

In my original definition, A_ZFC is concretely declared. I am sorry that I failed to check the translation.