Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

We study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

Nowhere precipitousness of some ideals

In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals on Pkλ, in particular the non-stationary ideal NSkλ under cardinal arithmetic assumptions.In this section I denotes a non-principal ideal on an infinite set A. Let I+ = PA / I (ordered by inclusion as a forcing notion) and I∣X = {Y ⊂ A: Y ⋂ X ∈ I}, which is also an ideal on A for X ∈ I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recall I is said to be precipitous if ⊨I+ “Ult(V, Ġ) is well-founded” [9].The central notion of this paper is a strong negation of precipitousness [1]:Definition. I is nowhere precipitous if I∣X is not precipitous for every X ∈ I+ i.e., ⊨I+ “Ult(V, Ġ) is ill-founded.”It is useful to characterize nowhere precipitousness in terms of infinite games (see [11, Section 27]). Consider the following game G(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately choose Xn ∈ I+ and Yn ∈ I+ respectively so that Xn ⊃ Yn ⊃n+1. After ω moves, Empty wins the game if⋂n<ωXn=⋂n<ωYn = Ø.See [5, Theorem 2] for a proof of the following characterization.Proposition. I is nowhere precipitous if and only if Empty has a winning strategy in G(I).

Saturated ideals and the singular cardinal hypothesis

The large cardinal-like properties of saturated ideals have been investigated by various authors, including Foreman [F], and Jech and Prikry [JP], among others. One of the most interesting consequences of a strongly compact cardinal is the following theorem of Solovay [So2]: if a strongly compact cardinal exists then the singular cardinal hypothesis holds above it. In this paper we discuss the question of relating the existence of saturated ideals and the singular cardinal hypothesis. We will show that the existence of “strongly” saturated ideals implies the singular cardinal hypothesis. As a biproduct we will present a proof of the above mentioned theorem of Solovay using generic ultrapowers. See Jech and Prikry [JP] for a nice exposition of generic ultrapowers. We owe a lot to the work of Foreman [F]. We would like to express our gratitude to Noa Goldring for many helpful comments and discussions.Throughout this paper we assume that κ is a strongly inaccessible cardinal and λ is a cardinal >κ. By an ideal on κλ we mean a κ-complete fine ideal on Pκλ. For I an ideal on κλ let PI denote the poset of I-positive subsets of κλ.Definition. Let I be an ideal on κλ. We say that I is a bounding ideal if 1 ⊩-PI “δ(δ is regular cardinal ”.We can show that if a normal ideal is “strongly” saturated then it is bounding.Theorem 1. If 1 is an η-saturated normal ideal onκλ, where η is a cardinal <λsuch that there are fewer thanκmany cardinals betweenκand η (i.e. η < κ+κ), then I is bounding.Proof. Let I be such an ideal on κλ. By the work of Foreman [F] and others, we know that every λ+-saturated normal ideal is precipitous. Suppose G is a generic filter for our PI. Let j: V → M be the corresponding generic elementary embedding. By a theorem of Foreman [F, Lemma 10], we know that Mλ ⊂ M in V[G]. By η-saturation, cofinalities ≥η are preserved; that is, if cfvα ≥ η, then cfvα = cfv[G]α. From j ↾ Vκ being the identity on Vκ and M being λ-closed in V[G], we conclude that cofinalities <κ are preserved. Therefore if cfvα ≠ cfv[G]α then κ ≤ cfvα < η.

A saturation property of ideals and weakly compact cardinals

Suppose κ is a regular uncountable cardinal, λ is a cardinal > 1, and is a κ-complete uniform ideal on κ. This paper deals with a saturation property Sat(κ, λ, ) of , which is a weakening of usual λ-saturatedness. Roughly speaking, Sat(κ, λ, ) means that can be densely extended to λ-saturated ideals on small fields of subsets of κ. We will show that some consequences of the existence of a λ-saturated ideal on κ follow from weaker ∃: Sat(κ, λ, ), and that ∃: Sat(κ, λ, ) is connected with weak compactness and complete ineffability of κ in much the same way as the existence of a saturated ideal on κ is connected with measurability of κ.In §2, we define Sat(κ, λ, ), mention a few results that can be proved by straightforward adaptation of known methods, and discuss generic ultrapowers of ZFC−, which will be used repeatedly in the subsequent sections as a main technical tool. A related concept Sat(κ, λ) is also defined and shown to be equivalent to ∃: Sat(κ, λ, ) under a certain condition.In §3, we show that ∃: Sat(κ, κ, ) implies that κ is highly Mahlo, improving results in [KT] and [So].

Weakly compact cardinals in models of set theory

The central notion of this paper is that of a κ-elementary end extension of a model of set theory. A model is said to be a κ-elementary end extension of a model of set theory if > and κ, which is a cardinal of , is end extended in the passage from to , i.e., enlarges κ without enlarging any of its members (see §0 for more detail). This notion was implicitly introduced by Scott in [Sco] and further studied by Keisler and Morley in [KM], Hutchinson in [H] and recently by the author in [E]. It is not hard to see that if has a κ-elementary end extension then κ must be regular in . Keisler and Morley [KM] noticed that this has a converse if is countable, i.e., if κ is a regular cardinal of a countable model then has a κ-elementary end extension. Later Hutchinson [H] refined this result by constructing κ-elementary end extensions 1 and 2 of an arbitrary countable model in which κ is a regular uncountable cardinal, such that 1 adds a least new element to κ while 2 adds no least new ordinal to κ. It is a folklore fact of model theory that the Keisler-Morley result gives soft and short proofs of countable compactness and abstract completeness (i.e. recursive enumera-bility of validities) of the logic L(Q), studied extensively in Keisler's [K2]; and Hutchinson's refinement does the same for stationary logic L(aa), studied by Barwise et al. in [BKM]. The proof of Keisler-Morley and that of Hutchinson make essential use of the countability of since they both rely on the Henkin-Orey omitting types theorem. As pointed out in [E, Theorem 2.12], one can prove these theorems using “generic” ultrapowers just utilizing the assumption of countability of the -power set of κ. The following result, appearing as Theorem 2.14 in [E], links the notion of κ-elementary end extension to that of measurability of κ. The proof using (b) is due to Matti Rubin.