On the ultrafilter of closed, unbounded sets

1979 ◽  
Vol 44 (4) ◽  
pp. 503-506
Author(s):  
D. A. Martin ◽  
W. Mitchell
Keyword(s):  
Measurable Cardinal ◽  
Turing Degree ◽  
Significant Part ◽  
Inner Models ◽  
Measurable Cardinals

Solovay proved in 1967 that the axiom of determinateness implies that the filter C generated by closed and unbounded subsets of ω1 is an ultrafilter. It has long been conjectured that a significant part of the theory of the axiom of determinateness should be provable from the hypothesis that C is an ultrafilter, but even the first step of finding inner models with several measurable cardinals has proved elusive. In this paper we show that such models exist. Much of our proof is a modification of Kunen's proof in [3] of the same conclusion from the existence of a measurable cardinal κ such that 2κ > κ+.Since no proof of Solovay's result seems to have been published, we insert a proof here. We want to show that for any set x ⊂ ω1 there is a closed, unbounded set either contained in or disjoint from x. By the lemma of [4] there is a Turing degree d such that either ω1e Є x for all degrees e ≥T d or ω1e ∉ x for all degrees e ≥T d. By a theorem of Sacks [1], [5] every d-admissible is ω1e for some e ≥T d, so it is enough to show that there is a closed, unbounded set of d-admissibles. Let a ⊂ ω have degree d; then is such a set.

Complexity of κ-ultrafilters and inner models with measurable cardinals

1984 ◽  
Vol 49 (3) ◽  
pp. 833-841 ◽  
Author(s):  
Claude Sureson
Keyword(s):  
Regular Cardinal ◽  
Measurable Cardinal ◽  
Order Type ◽  
Inaccessible Cardinal ◽  
Inner Models ◽  
Inner Model ◽  
Measurable Cardinals

The purpose of this paper is to establish a connection between the complexity of κ-ultrafilters over a measurable cardinal κ, and the existence of ascending Rudin-Keisler chains of κ-ultrafilters and of inner models with several measurable cardinals.If V is a model of ZFC + “There exists a measurable cardinal κ”, then V satisfies “There exists a normal κ-ultrafilter”, that is to say a “simple” κ-ultrafilter. The only known examples of “complex” κ-ultrafilters have been constructed by Kanamori [2], Ketonen [4] and Kunen (cf. [2]) with stronger hypotheses than measurability: compactness or supercompactness. Using the notions of skies and constellations defined by Kanamori [2] for the measurable case, and which witness the complexity of a κ-ultrafilter, we shall show the necessity of such assumptions, namely:Theorem 1. If λ < κ is a strongly inaccessible cardinal, the existence of a κ-ultrafilter with more than λ constellations implies that there is an inner model with two measurable cardinals if λ = ω and λ + 1 measurable cardinals otherwise.Theorem 2. Let θ < κ be an arbitrary ordinal. If there is a κ-ultrafilter such that the order-type of its skies is greater than ωθ, then there exists an inner model with θ + 1 measurable cardinals.And as a corollary, we obtain:Theorem 3. Let μ < κ be a regular cardinal. If there exists a κ-ultrafilter containing the closed-unbounded subsets of κ and {α < κ: cf(α) = μ}, then there is an inner model with two measurable cardinals if μ = ω, and μ + 1 measurable cardinals otherwise.

Forcing the failure of CH by adding a real

1984 ◽  
Vol 49 (4) ◽  
pp. 1185-1189 ◽  
Author(s):  
Saharon Shelah ◽  
Hugh Woodin
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Inaccessible Cardinal ◽  
Inner Models ◽  
Inner Model ◽  
Independence Results ◽  
Measurable Cardinals

We prove several independence results relevant to an old question in the folklore of set theory. These results complement those in [Sh, Chapter XIII, §4]. The question is the following. Suppose V ⊨ “ZFC + CH” and r is a real not in V. Must V[r] ⊨ CH? To avoid trivialities assume = .We answer this question negatively. Specifically we find pairs of models (W, V) such that W ⊨ ZFC + CH, V = W[r], r a real, = and V ⊨ ¬CH. Actually we find a spectrum of such pairs using ZFC up to “ZFC + there exist measurable cardinals”. Basically the nicer the pair is as a solution, the more we need to assume in order to construct it.The relevant results in [Sh, Chapter XIII] state that if a pair (of inner models) (W, V) satisfies (1) and (2) then there is an inaccessible cardinal in L; if in addition V ⊨ 2ℵ0 > ℵ2 then 0# exists; and finally if (W, V) satisfies (1), (2) and (3) with V ⊨ 2ℵ0 > ℵω, then there is an inner model with a measurable cardinal.Definition 1. For a pair (W, V) we shall consider the following conditions:(1) V = W[r], r a real, = , W ⊨ ZFC + CH but CH fails in V.(2) W ⊨ GCH.(3) W and V have the same cardinals.

An AD-like model

1985 ◽  
Vol 50 (2) ◽  
pp. 531-543 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Strong Sense ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Partial Orderings ◽  
Measurable Cardinals ◽  
Jonsson Cardinals

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].Another such model will be constructed in this paper. Specifically, the following theorem is proven.Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.

On P-points over a measurable cardinal

1981 ◽  
Vol 46 (1) ◽  
pp. 59-66
Author(s):  
A. Kanamori
Keyword(s):  
Measurable Cardinal ◽  
Focus Of Attention ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Inner Model ◽  
Measurable Cardinals ◽  
The Universe

This paper continues the study of κ-ultrafilters over a measurable cardinal κ, following the sequence of papers Ketonen [2], Kanamori [1] and Menas [4]. Much of the concern will be with p-point κ-ultrafilters, which have become a focus of attention because they epitomize situations of further complexity beyond the better understood cases, normal and product κ-ultrafilters.For any κ-ultrafilter D, let iD: V → MD ≃ Vκ/D be the elementary embedding of the universe into the transitization of the ultrapower by D. Situations of U < RKD will be exhibited when iU(κ) < iD(κ), and when iU(κ) = iD(κ). The main result will then be that if the latter case obtains, then there is an inner model with two measurable cardinals. (As will be pointed out, this formulation is due to Kunen, and improves on an earlier version of the author.) Incidentally, a similar conclusion will also follow from the assertion that there is an ascending Rudin-Keisler chain of κ-ultrafilters of length ω + 1. The interest in these results lies in the derivability of a substantial large cardinal assertion from plausible hypotheses on κ-ultrafilters.

Nonabsoluteness of elementary embeddings

1989 ◽  
Vol 54 (3) ◽  
pp. 774-778
Author(s):  
Friedrich Wehrung
Keyword(s):  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Minimal Support ◽  
Elementary Embeddings ◽  
Measurable Cardinals

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU(V). Ifxis a set, say thatUmovesxwhenjU(x)≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um(m, n ∈ ω). Then, for each θ inUltU(V),if E is the (minimal) support of θ inUltU(V),then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω(m + 1))≠ ∅.

Minimality in the Δ⅓-degrees

1987 ◽  
Vol 52 (4) ◽  
pp. 908-915
Author(s):  
Philip Welch
Keyword(s):  
Upper Bound ◽  
Countable Collection ◽  
Inner Models ◽  
Measurable Cardinals

AbstractWe show in ZFC, assuming all reals have sharps, that a countable collection of Δ⅓-degrees without a minimal upper bound implies the existence of inner models with measurable cardinals.

The measure quantifier

1979 ◽  
Vol 44 (1) ◽  
pp. 103-108 ◽  
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Measurable Cardinal ◽  
Order Logic ◽  
First Order Logic ◽  
Order Structure ◽  
Proper Class ◽  
First Order ◽  
Recursive Enumerability ◽  
Almost All ◽  
Measurable Cardinals ◽  
The Universe

Originally generalized quantifiers were introduced to specify that a given formula was true for “many x's” e.g. ⊨ Qxφ(x) iff card{x ∈ ∣∣ ∣ ⊨ φ[x]} ≥ ℵ0, ℵ1, or some fixed cardinal κ. In this paper we formalize the notion that φ{x) is true “for almost all x”. This is accomplished by referring to structures = (′, μ) where ′ is a first-order structure and μ is a measure of a suitable type on the universe of ′. We will prove that the language Lμ obtained from first-order logic by adjoining a quantifier Qμ, which ranges over the measure μ, is fully compact if we assume the existence of a proper class of measurable cardinals. As a corollary to the compactness theorem we obtain the recursive enumerability of the validities of Lμ. Finally, the Magidor-Malitz quantifiers Qkn (n ∈ ω) will be added to Lμ together with analogous quantifiers Qμm (m ∈ ω) to form Lκμ<ω,<ω, which is compact for sets of sentences of cardinality < κ, where κ is a measurable cardinal > ℵ0.An alternate approach to formalizing “for almost all” has been recently developed by Barwise, Kaufmann and Makkai [1] who follow a suggestion of Shelah [5].

A measurable cardinal with a nonwellfounded ultrapower

1980 ◽  
Vol 45 (3) ◽  
pp. 623-628 ◽  
Author(s):  
Mitchell Spector
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Axiom Of Choice ◽  
Partial Ordering ◽  
Partition Relation ◽  
Partition Relations ◽  
Consistency Results ◽  
The Axiom Of Choice ◽  
Measurable Cardinals ◽  
Weakened Form

The usefulness of measurable cardinals in set theory arises in good part from the fact that an ultraproduct of wellfounded structures by a countably complete ultrafilter is wellfounded. In the standard proof of the wellfoundedness of such an ultraproduct, one first shows, without any use of the axiom of choice, that the ultraproduct contains no infinite descending chains. One then completes the proof by noting that, assuming the axiom of choice, any partial ordering with no infinite descending chain is wellfounded. In fact, the axiom of dependent choices (a weakened form of the axiom of choice) suffices. It is therefore of interest to ask whether some use of the axiom of choice is needed in order to prove the wellfoundedness of such ultraproducts or whether, on the other hand, their wellfoundedness can be proved in ZF alone. In Theorem 1, we show that the axiom of choice is needed for the proof (assuming the consistency of a strong partition relation). Theorem 1 also contains some related consistency results concerning infinite exponent partition relations. We then use Theorem 1 to show how to change the cofinality of a cardinal κ satisfying certain partition relations to any regular cardinal less than κ, while introducing no new bounded subsets of κ. This generalizes a theorem of Prikry [5].

The consistency strength of the free-subset property for ωω

1984 ◽  
Vol 49 (4) ◽  
pp. 1198-1204 ◽  
Author(s):  
Peter Koepke
Keyword(s):  
Measurable Cardinal ◽  
Core Model ◽  
Reverse Direction ◽  
Ground Model ◽  
Consistency Strength ◽  
The Core ◽  
Inner Model ◽  
Free Subset ◽  
Coherent Sequence ◽  
Measurable Cardinals

A subset X of a structure S is called free in S if ∀x ∈ Xx ∉ S[X − {x}]; here, S[Y] is the substructure of S generated from Y by the functions of S. For κ, λ, μ cardinals, let Frμ(κ, λ) be the assertion:for every structure S with κ ⊂ S which has at most μ functions and relations there is a subset X ⊂ κ free in S of cardinality ≥ λ.We show that Frω(ωω, ω), the free-subset property for ωω, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].In the first section of this paper we prove some combinatorial facts about Frμ(κ, λ); in particular the first cardinal κ such that Frω(κ, ω) is weakly inaccessible or of cofinality ω (1.2). The second section shows that, under Frω(ωω, ω), ωω is measurable in an inner model. For the convenience of readers not acquainted with the core model κ, we first deduce the existence of 0# (2.1) using the inner model L. Then we adapt the proof to the core model and obtain that ωω is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Frω(ωω, ω) over a ground model which contains an ω-sequence of measurable cardinals. We show in §4 that indeed a coherent sequence of Ramsey cardinals suffices. In §3 we obtain such a sequence as an endsegment of a Prikry sequence.

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