User blog comment:P進大好きbot/Whether Rayo's number is well-defined or not/@comment-25601061-20180606221005/@comment-35470197-20181027124512

@Emlightened

Oops. I did not notice your reply. Sorry for being too late.

> And you mention "A maximal consistent set" (alternative definition section, first bullet) without specifying which - this is my gripe.

What you wrote is that there is no maximal consistent set, but not is that there is no definable/specified maximal consistent set. So you are wrong. There IS a maximal consistent set, which is unspecified.

In this section, I gave an alternative direction, which is not based on ZFC set theory. Therefore it causes no problem if I add a new constant term symbol corresponding to \(\Sigma\) and an axiom which ensures that it is a maximal consistent set. As I wrote, it is not a definition under ZFC.

> My interpretation of how Rayo's Number is defined: > > Why doesn't this work? And moreover, why are all of your theories set theories, as opposed to including theories about \(\mathbb N\)?

I have never said "there is no way to define Rayo number under any formal system". In this blog post, I just mentioned that ZFC set theory is not sufficient for this puropose. For example, it is not so difficult to formalise the definition under specific axioms and semantics of second order logic or even first order MK set theory.

Ok, I add the description in order to make it clearer.