User blog comment:P進大好きbot/Whether Rayo's number is well-defined or not/@comment-11227630-20181205112602/@comment-35470197-20181205141040

You are referring to my alternative definition of Rayo's number using a maximal consistent set \(\Sigma\) of formulae, right? Then more precisely, we can consider the alternative Rayo's number corresponding to PA or KP in ZFC set theory (not PA or KP). (When you consider PA, then replace \(\Phi^V\) by \(\Phi^{\mathbb{N}}\) so that the satisfaction makes sense.)

If you replace the occurence of ZFC by other (weaker, stronger, or irrelevant) axioms, then the resulting Rayo's number can be greater, and also can be smaller depending on the corresponding choice of \(\Sigma\).

Say, let \(\Sigma_A\) denote a choice of such a (non-unique) \(\Sigma\) for a given axiom \(A\). If \(\Sigma_{\textrm{KP}} = \Sigma_{\textrm{ZFC}}\), then they share the values of the resulting Rayo's number. If \(\Sigma_{\textrm{PS}}\) coincides with the subset of \(\Sigma_{\textrm{ZFC}}\) consisting of arithmetic formulae, then the Rayo's number corresponding to PA is smaller than the Rayo's number correspnding to ZFC, because the former one is only allowed to use the truth predicate at arithmetic formulae. If there are not such relations on \(\Sigma\)'s, then we do not have a similar comparison.