User blog comment:Nayuta Ito/Is this bigger than Rayo's number?/@comment-27513631-20180504234554/@comment-27513631-20180505082741

Ooh, thank you for the clarifications.

The usually accepted definitions of largest numbers over here, unfortunately, instead of defining relative to a model, are defined relative to some absolute platonist universe which satisfies the axioms of ZFC and whatever else the reader thinks is true. Personally, I don't like this, as it forces us to use a specific philosophy.

Now, because of this, it's very hard to make numbers in any way other than in a Rayo-like construction. In particular, suppose we were to use the internal language only. We can define \(R_r(n)\) to be the largest natural number \(h\) such that, for some \(\Pi_r\) formula \(\phi\) with quantifier depth \(\leq n\) and only free variable \(x\), \(\forall x(\phi(x) \leftrightarrow x=h)\).

This means that, if \(f\) is any \(\Pi_r\) definable function (an all definable \(f\) are \(\Pi_r\) for some \(r\)), \(R_r\) eventually dominates it. Moreover, \(R_r\) is internally definable.

Now, the reason why I'm strongly inclined to believe that it's small, compared even to the internally well-defined R_{9}(9^9), is that there is no reference to truth, just provability. Specifically, you never seem to consider whether a statement is true in the universe, but only whether it is provable in the metatheory.

Now, suppose that there is some internal, definable \(g\), which is in \(\mathbb N^{\mathbb N}\), but not provably so. Does your function outgrow it? Some \(R_r\) does, but I'm not convinced your function gets anywhere near, since wouldn't it require that you can prove that \(g\) is in \(\mathbb N^{\mathbb N}\)? Can you explain this to me, in case I've made mistakes?

There is a way in which the meta-naturals correspond to the internal naturals of any theory - namely, for each meta-natural n we apply the internal successor n times to the internal 0. We can't go the other way, though. I believe that due to the platonistic interpretation, we work with a ω-model anyway, so this isn't something taken into consideration.

"However, A must satisfy these conditions: A connotes Peano arithmetic;" still not understanding this bit sorry - surely this is both impossible (PA is not finitely axiomisable in FOL) and determines A up to uniqueness.

Having to work in a prespecified base meta-theory sounds like a pain. Hope it works out better next time.