Generic Ultrapowers Research Articles

Forcing revisited

AbstractThe purpose of this paper is to propose and explore a general framework within which a wide variety of model construction techniques from contemporary set theory can be subsumed. Taking our inspiration from presheaf constructions in category theory and Boolean ultrapowers, we will show that generic extensions, ultrapowers, extenders and generic ultrapowers can be construed as examples of a single model construction technique. In particular, we will show that Łoś's theorem can be construed as a specific case of Cohen's truth lemma, and we isolate the weakest conditions a filter must satisfy in order for the truth lemma to work.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

AbstractWe 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.

Martin’s Maximum and tower forcing

There are several examples in the literature showing that compactness-like properties of a cardinal κ cause poor behavior of some generic ultrapowers which have critical point κ (Burke [1] when κ is a supercompact cardinal; Foreman-Magidor [6] when κ = ω 2 in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $$\overrightarrow I $$ is a tower of ideals which concentrates on the class $$GI{C_{{\omega _1}}}$$ of ω 1-guessing, internally club sets, then $$\overrightarrow I $$ is not presaturated (a set is ω 1-guessing iff its transitive collapse has the ω 1-approximation property as defined in Hamkins [10]). This theorem, combined with work from [16], shows that if PFA + or MM holds and there is an inaccessible cardinal, then there is a tower with critical point ω 2 which is not presaturated; moreover, this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor [6]) to exist in all models of Martin’s Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω 2 has similar implications for towers of ideals which concentrate on the wider class $$GI{C_{{\omega _1}}}$$ of ω 1-guessing, internally stationary sets. Finally, we show that the word “presaturated” cannot be replaced by “precipitous” in the theorems above: Martin’s Maximum (which implies SRP and the Tree Property at ω 2) is consistent with a precipitous tower on $$GI{C_{{\omega _1}}}$$ .

On almost precipitous ideals

With less than 0# two generic extensions ofL are identified: one in which \({\aleph_1}\), and the other \({\aleph_2}\), is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in Gitik M, Magidor M (On partialy wellfounded generic ultrapowers, in Pillars of Computer Science, 2010), and answers some questions raised there. Also, main results of Gitik (On normal precipitous ideals, 2010), are generalized—assumptions on precipitousness are replaced by those on ∞-semi precipitousness. As an application it is shown that if δ is a Woodin cardinal and there is an \({f:\omega_1 \to \omega_1}\) with \({\|f\|=\omega_2}\), then after \({Col(\aleph_2,\delta)}\) there is a normal precipitous ideal over \({\aleph_1}\). The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal.

Generalized Prikry forcing and iteration of generic ultrapowers

AbstractIt is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n(κ) | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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 onPkλ, in particular the non-stationary idealNSkλunder cardinal arithmetic assumptions.In this sectionIdenotes a non-principal ideal on an infinite setA. LetI+=PA/I(ordered by inclusion as a forcing notion) andI∣X= {Y⊂A:Y⋂X∈I}, which is also an ideal onAforX∈I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recallIis 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.Iis nowhere precipitous ifI∣Xis not precipitous for everyX∈ 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 gameG(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately chooseXn∈I+andYn∈I+respectively so thatXn⊃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.

Menas' conjecture and generic ultrapowers

We apply the technique of generic ultrapowers to study the splitting problem of stationary subsets of P Kλ . We present some conditions which guarantee the splitting of stationary subsets of P Kλ .

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].

Certain Values of Completeness and Saturatedness of a Uniform Ideal Rule out Certain Sizes of The Underlying Index Set

AbstractUsing the method of non-well-founded generic ultra-powers, we shall prove a generalization of a theorem of Taylor that certain values of completeness and saturatedness of a uniform ideal rule out certain sizes of the underlying index set.

Thomas Jech and Karel Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593–595. - F. Galvin, T. Jech, and M. Magidor. An ideal game. The journal of symbolic logic, vol. 43 (1978), pp. 284–292. - T. Jech, M. Magidor, W. Mitchell, and K. Prikry. Precipitous ideals. The journal

Thomas Jech and Karel Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593–595. - F. Galvin, T. Jech, and M. Magidor. An ideal game. The journal of symbolic logic, vol. 43 (1978), pp. 284–292. - T. Jech, M. Magidor, W. Mitchell, and K. Prikry. Precipitous ideals. The journal

Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers

Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers