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.

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.

INNER MODEL THEORETIC GEOLOGY

2016 ◽  
Vol 81 (3) ◽  
pp. 972-996 ◽  
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Set Theory ◽  
Main Theme ◽  
Solid Core ◽  
Strong Cardinal ◽  
The Third ◽  
Inner Models ◽  
Inner Model ◽  
Basic Concepts ◽  
Woodin Cardinal ◽  
The Universe

AbstractOne of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

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.

Natural internal forcing schemata extending ZFC: Truth in the universe?

1994 ◽  
Vol 59 (2) ◽  
pp. 461-472
Author(s):  
Garvin Melles
Keyword(s):  
Set Theory ◽  
Mathematical Truth ◽  
Unified Theory ◽  
Rule Of Thumb ◽  
Guiding Principle ◽  
Survey Article ◽  
Inner Models ◽  
Independence Results ◽  
Universe Of Sets ◽  
The Universe

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his article What is Maclane Missing? challenging Mac Lane's view of set theory.

Inner models for set theory—Part I

1951 ◽  
Vol 16 (3) ◽  
pp. 161-190 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Large Family ◽  
Axiom System ◽  
Axiom Of Choice ◽  
Axiomatic System ◽  
Additional Hypothesis ◽  
Inner Models ◽  
Inner Model ◽  
The Axiom Of Choice

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the basic axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann (4), Mostowski (5), and more recently Gödel (1), who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model.

On the ultrafilters and ultrapowers of strong partition cardinals

1984 ◽  
Vol 49 (4) ◽  
pp. 1268-1272
Author(s):  
J.M. Henle ◽  
E.M. Kleinberg ◽  
R.J. Watro
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Partition Property ◽  
Axiom Of Determinacy ◽  
Almost Everywhere ◽  
Normal Measure ◽  
Constructible Sets ◽  
Inner Model ◽  
The Axiom Of Choice ◽  
Strong Partition Cardinals

A strong partition cardinal is an uncountable well-ordered cardinal κ such that every partition of [κ]κ (the size κ subsets of κ) into less than κ many pieces has a homogeneous set of size κ. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including ℵ1 have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal κ is measurable; in fact every stationary subset of κ is measure one under some normal measure on κ. If μ is a countably additive ultrafilter extending the closed unbounded filter on κ, then the length of the ultrapower [κ]κ under the less than almost everywhere μ ordering is again a measurable cardinal. In §1 below we establish a polarized partition property on these measurable cardinals.

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.

scholarly journals Internal Consistency and the Inner Model Hypothesis

2006 ◽  
Vol 12 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Sy-David Friedman
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Weak Sense ◽  
Large Cardinals ◽  
Model Method ◽  
Inner Models ◽  
Inner Model ◽  
The Relationship ◽  
The Universe ◽  
The Way

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening of Cohen's result:Theorem 1 (Easton's Theorem). There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals.

Co-critical points of elementary embeddings

1985 ◽  
Vol 50 (1) ◽  
pp. 220-226
Author(s):  
Michael Sheard
Keyword(s):  
Critical Point ◽  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Ground Model ◽  
Target Model ◽  
Inner Models ◽  
Inner Model ◽  
Elementary Embeddings

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model” (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinalκ, then the ultraproduct of the ground model viacollapsesκ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.Definition 1.1. LetMandNbe inner models (transitive, proper class models) of ZFC, and letj:M→Nbe an elementary embedding. Theco-critical pointofjis the least ordinalλ, if any exist, such that there isX⊆λ, X∈NbutX∉M. Such anXis called anew subsetofλ.It is easy to see that the co-critical point ofj:M→Nis a cardinal inN.

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