Logical limit laws and analytic combinatorics for the proof-theoretic ordinal of PA

Given a sentence S from the first order (or monadic second order) language of linear orders one might ask what the probability is that S is true on the linear order determined by a randomly chosen ordinal below the proof-theoretic ordinal of PA. Here the notion of randomness is modeled in terms of asymptotic density or averaged asymptotic density. These densities are tied to counting questions for ordinals which are amenable to analytic combinatorics.

In this talk we will discuss some older results which have been obtained jointly with Alan Woods and we also cover a recent averaged zero one law emerging from the standard Gödel coding which is based on prime factorization and coding subterms at exponential positions. To prove the more recent results we used modern AI tools to some extent.