User blog comment:Nayuta Ito/Is this bigger than Rayo's number?/@comment-27513631-20180504234554/@comment-35470197-20180505104516

> there is no reference to truth, just provability.

I disagree.

I assumed that Con(A_ZFC) implies Con(A), which is not necessarily provable in the meta theory.

Also, I assumed that "80(P[x/t] \wedge 16x((P)48(x=t)))" i.e. "\neg(P[x/t] \wedge \exists x(P \to (x=t)))" is NOT provable under A. If a model satisfies a proposition, it is not unprovable. So the assumption is weaker than that of truth.

> There is a way in which the meta-naturals correspond to the internal naturals of any theory

Right, if you work with such a thoery. I pointed out it because in the original definition of Rayo's number, there is no declaration of axioms and the notion of natural numbers in it. In this case, one have to fix a way to correspond meta-naturals to naturals in non-canonical way. Since you are working on a model of ZFC, it is ok.

>  (PA is not finitely axiomisable in FOL)

YES. See the procise definition of FOL again. Maybe you are refering to FOL', but not to FOL. FOL is not a formal language of a first order logic.

> and determines A up to uniqueness.

I meant that I fixes an embedding so that terms (finitely many successors applied to 0) in PA makes sense in A. It is just a way to define the notion of natural numbers in A.