User blog:TheKing44/The analytical beaver functions

Rayo's function is based on the language of first order set theory, which is rather powerful. I think it would be interesting to take a step back to a weaker language: second order arithmetic.

Let $$\mathcal N$$ be the standard model of second order arithmetic. We say that a formula $$\phi$$ that takes one argument defines a natural number $$k$$ if $$\forall j \in \mathbb N. (j = k \iff \mathcal N \models \phi(j))$$. That is, $$k$$ is the unique natural number that satisfies $$\phi$$ in the standard model.

We define the analytical beaver function $$\operatorname{AnaB}_{\Delta^2_0}(n)$$ to be the the largest natural number definable by a formula that is less than $$n$$ symbols long plus 1 (or 0 if there is no such number). Since there are only finitely many such formulas, and each formula defines at most one number, this is well defined. As to define a concrete number, we will define the analytical beaver number to be $$\operatorname{AnaB}_{\Delta^2_0}(10^{100})$$.

Now, you may be wondering what $$\Delta^2_0$$ is doing there? Well, that symbol denotes the set of all formulas in the language of second order arithmetic. In general, we define $$\operatorname{AnaB}_S(n)$$ to be the largest natural number definable definable by a formula in the set $$S$$ that is less than $$n$$ symbols plus 1 (or 0 is there is no such number). Some interesting sets of formulas are those in the analytical hierarchy (which includes the arithmetical hierarchy). For example, $$\operatorname{AnaB}_{\Delta^0_1}$$ would grow about as fast as the traditional busy beaver function.

Question: It is clear that $$\operatorname{AnaB}_{\Delta^2_0}$$ is not in the analytical hierarchy itself. However, $$\operatorname{AnaB}_S$$ is, for any other $$S$$ in the analytical hierarchy. Where do such $$\operatorname{AnaB}_S$$ fall in the hierarchy? We know they would not fall in $$S$$, for example.