An application of Σ40 determinacy to the degrees of unsolvability

Abstract

If a and b are degrees of unsolvability, a is called (as in [1]) a minimal cover of b if b < a and no degree c satisfies b < c < a. A degree a is called a minimal cover if it is a minimal cover of some degree b. We show in ZF set theory that there is a degree d such that every degree a ≥ d is a minimal cover. In [1] it was remarked that this result follows via a lemma of D. A. Martin [4] from the determinacy of a certain Σ50 Gale-Stewart game (which in turn follows [5] from the existence of a measurable cardinal). The argument here is parallel to that of [1], but the essential new ingredient is the result of Paris [6] that Σ50 determinacy can be proved in ZF. Also it is necessary to replace the Σ50 game of [1] by a related Σ40 game and to use the method, rather than just the statement, of Martin's lemma.Theorem. There is a degreedsuch that every degreea ≥ dis a minimal cover.Proof. In a (Gale–Stewart) game on 2ω, players I and II construct a set of integers A as follows: At the nth stage, player I determines whether n ϵ A if n is even and player II if n is odd. Thus player I controls A0 and player II controls A1 where Ai = {n: 2n + i ϵ A}. Let Aij denote (Ai)j and let M (A, B) mean that the degree of A is a minimal cover of that of B. Consider now the particular game in which player I wins iff M(A, A00) holds.