An undecidable extension of Morley's theorem on the number of countable models

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of $\sigma$-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.Comment: 31 page

Proper forcings and absoluteness in L(R)L(\Bbb R)

We show that in the presence of large cardinals proper forcings do not change the theory of ${L({\Bbb R})}$ with real and ordinal parameters and do not code any set of ordinals into the reals unless that set has already been so coded in the ground model

Set theory and the analyst

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure - category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know"

Homogeneously Souslin sets in small inner models

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0 long 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 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 ω), (πst: s, t ∈ <ω ω, s ⊂ t)) such that M0 = V, each Ms is an inner model of ZFC with α Ms ⊂ Ms, each πst: Ms → Mt is an elementary embedding, and if x ∈ ω ω and (Mx, (πx↾n,x: n < ω)) is the direct limit of ((Mx↾n: n < ω), (πx↾n,x↾m: 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 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 2 ℵ0-closed embedding normal form then A is homogeneously Sousli