User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-44466522-20191230164711/@comment-35470197-20191230231243

Right, given an uncomputable large number n, the value of the relativised number M^n ∈ M heavily depends on a transitive model M. It does not mean that n is ill-defined, though. The value of n itself is not relativised, i.e. defined in V, which is unique.

I guess that you care the situation that n is defined by a truth predicate on M. In this case, n is defined in M, and hence the dependency on M is critical when M is unspecified. On the other hand, Rayo's number is intended to use a truth predicate on V, which can be formalised in some second order set theory. Therefore it is not the case.

As a related fact, Rayo's number is not naively formalised in ZFC set theory, because the use of truth predicate is restricted to a transitive small model M, but there is no specific M definable in ZFC as long as it is consistent. So if we stand on the position where we work in ZFC as long as no axiom is specified, then it is ill-defined. (Rayo just stated that he works in second order logic, and did not specify axioms.) For more details on this issue on the ill-definedness of a first order formalisation of Rayo's number, see this.