User blog comment:Primussupremus/What are the limits of Googology?/@comment-30754445-20170531225910/@comment-27513631-20170601150655

Have to disagree with your edit there, Psi. I think that to formalise Rayo's number inside a model \(M\), we need to work outside of \(M\), not inside, as our \(f\) is \(f:\omega^M \to \omega^M\), but may not be \(\in M\), due to not being \(\Pi_n\)-definable for any \(n\) (as it may involve \(\Sigma_{n+1}\) formulae, for instance), and hence not formalizable inside the model.

However, everything becomes a lot easier if we work outside the model. Here, we are essentially asserting the existence of a function \(\text{Rayo}:\omega^M \to \omega^M\) such that, for each \(n\in\omega^M\), there is a \((<_\mathbb{N})^M\)-maximal element of some \(M\)-finite set \(A\cap\omega^M\). Here, \(A\) is the set of unique values \(\in M\) produced by one of the short enough \(M\)-formulae.

To me, the mathematical issue would be asserting that there is exactly one function \(\text{Rayo}\) satisfying that property, and that the above (albeit slightly loose) definition captures the definition of \(\text{Rayo}\). I don't think I've missed anything out, and it seems like it wouldn't be difficult for a model theorist to prove exists, although there is no reason to believe that \(\text{Rayo} \in M\).

The philosophical issues are more complex. The above definition would be fine to a formalist: every mathematical 'universe' is just a model, whatever that is. However, the platonists would disagree, and say that Rayo's function can only exist over arbitrary models, but possibly not the actual universe. As a non-platonist, I refuse to hazard a guess as to how they'd justify this construction's existence in the Universe, but that seems to be mainly philosophy, to me.