User blog comment:DontDrinkH20/H-Boogol-Boogol: Hopefully a very big non-salad number/@comment-35470197-20180819021045

> There exists exactly one \(x∈N\) such that (N \models x\)

Forgetting \(\varphi_0\).

> FINALLY, let \(\mathcal{H}_a(b)\) be the smallest number of successor functions needed to describe \(\mathcal{H}_a(b)\) in \(M\) for all class-sized models \(\mathcal{M}\) of ZF.

Typo? The object \(\mathcal{H}_a(b)\) is multiply-defined.

Moreover, how do you define the smallest number using the quantification of class-sized (i.e. unbounded) models of ZF? Although it is invalid in the usual set theory like ZFC or NBG, is it valid in model theory?

Here, a class-sized model means a definable class which is proof-theoretically satisfies ZF, right? Since class-sized models are not formalised in a way similar to set-sized models, the satisfaction is very weakened here.

Also, could you tell me the assumption on the well-definedness of your number? (e.g. the consistency of ZF or something like that)

> Proof 4

The number \(\mu(\textrm{ZFC})\) is not well-defined in your way because \(\mathcal{L}(\textrm{ZFC})\) is not finite. Recall that \(\textrm{ZFC}\) admits an axiom schema which contains infinitely many symbols. You need to consider a finite segment of ZFC, which is sufficient to define the Rayo's number.