Arithmetical transfinite induction

and hierarchies of functions by Zygmunt Ra t a j c z yk (Warszawa) Abstract. We generalize to the case of arithmetical transfinite induction the follow-ing three theorems for PA: the Wainer Theorem, the Paris–Harrington Theorem, and a version of the Solovay–Ketonen Theorem. We give uniform proofs using combinatorial constructions. 1. Introduction. Th