Ramsey's Theorem does not Hold in Recursive Set Theory

Abstract

This chapter discusses Ramsey's theorem that does not hold in recursive set theory. The theorem is described, which states that there exists a recursive binary symmetric relation (R) on natural numbers (N) such that no recursively enumerable infinit subset of N is R-homogeneous. The proof of the theorem is based on the existence of two recursively enumerable sets of incomparable degrees of insolvability. The proofs of Ramsey's theorem show that there exist arithmetical R-homogeneous sets for recursive relations R. The existence of an infinite recursively enumerable R -homogeneous set implies that either S 1 is recursive in S 2 or S 2 is recursive in S 1 . There exist recursively enumerable sets of incomparable degrees.