Homogeneously Souslin sets in small inner models

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0long does not exist, or else (b) V = K, where K is the core model below a μ-measurable cardinal. 1 Homogeneously Souslin sets. In this paper we shall deal with homogeneously Souslin sets of reals, or rather with sets of reals which admit an ω-closed embedding normal form. Definition 1.1 (Cf. [4, p. 92].) Let A ⊂ ω. Let α ∈ OR. We say that A has an α-closed embedding normal form if and only if the following holds true. There is a commutative system ((Ms: s ∈ ω), (πst: s, t ∈ ω, s ⊂ t)) such that M0 = V , each Ms is an inner model of ZFC with Ms ⊂ Ms, each πst: Ms → Mt is an elementary embedding, and if x ∈ ω and (Mx, (πx1n,x: n < ω)) is the direct limit of ((Mx1n: n < ω), (πx1n,x1m: n ≤ m < ω)) then x ∈ A ⇔ Mx is wellfounded. As we shall not need it here, we do not repeat the definition of the concept of being homogeneously Souslin in this paper (cf. [4, p. 87]). We just remind the reader of the following facts. Lemma 1.2 Let A ⊂ ω. (1) If A is coanalytic and if κ is a measurable cardinal then A is κ-homogeneously Souslin (cf. [3], [4, Theorem 2.2]). (2) If A is κ-homogeneously Souslin, where κ is a (measurable) cardinal, then A is determined (cf. [4, Theorem 2.3]) and has a κ-closed embedding normal form (cf. [4, p. 92]). (3) If A has a 2א0-closed embedding normal form then A is homogeneously Souslin (cf. [7, Lemma 2.5], [2, Theorem 5.2]).