User blog comment:Ubersketch/Does the strength of a collection of axioms increase the growth rate of it's corresponding Rayo?/@comment-39605890-20190609052643/@comment-35470197-20190609062204

Since Rayo used second order logic to formalise truth predicate at \(V\), I do not think that he intended to use the satisfaction at models. If he had intended so, then he would heve used an well-ordering on \(V\), e.g. \(V = L\), and the consistency of \(\textrm{ZFC}\) so that a model is definable, but he had never declared such axioms.

I guess that what you are talking is not the orginal definition in second order logic, but an interpretation in first order logic using models.