A Footnote to a Theorem of Solovay on Recursive Encodability

Abstract

This chapter presents a new proof of Solovay's effective Ramsey theorem, which states that every recursively encodable set is hyperarithmetic. This is used as a lemma to prove an even more effective Ramsey theorem and proceed to the main theorem by an easy stage construction. The proof presented in the chapter directly follows Solovay for many key ideas and individual steps. A set of natural numbers is recursively encodable if it is recursive in some subset of every infinite set. For s and X sets of natural numbers with X infinite and P a set of sets, the pair 〈s,X〉 lands in P if for every infinite Y ⊆ X, s ∪ Y∈ P; it avoids P if it lands in the complement of P. X itself lands in P if 〈0,X〉 does so; likewise for avoids. The Nash–Williams strengthening of Ramsey's theorem states that if P is open, there is an infinite set that either lands in P or avoids P.