scholarly journals Determinacy in strong cardinal models

2011 ◽  
Vol 76 (2) ◽  
pp. 719-728
Author(s):  
P. D. Welch
Keyword(s):  
Strong Cardinal ◽  
Inner Models ◽  
Inner Model ◽  
Strong Cardinals

AbstractWe give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:Theorem A. Det(-IND) ⇒ there exists an inner model with a strong cardinal.Theorem B. Det(AQI) ⇒ there exist type-l mice and hence inner models with proper classes of strong cardinals.where -IND(AQI) is the pointclass of boldface -inductive (respectively arithmetically quasi-inductive) sets of reals.

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.

scholarly journals Laver sequences for extendible and super-almost-huge cardinals

1999 ◽  
Vol 64 (3) ◽  
pp. 963-983 ◽  
Author(s):  
Paul Corazza
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Large Cardinal ◽  
Strong Cardinal ◽  
Regular Class ◽  
Strong Cardinals ◽  
Extendible Cardinals

AbstractVersions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses. Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals.

0# and inner models

2002 ◽  
Vol 67 (3) ◽  
pp. 924-932 ◽  
Author(s):  
SY D. Friedman
Keyword(s):  
High Order ◽  
Covering Theorem ◽  
Inner Models ◽  
Inner Model ◽  
Mahlo Cardinals ◽  
Reverse Easton Iteration

In this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0#. We show, assuming that 0# exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal κ which is κ-Mahlo. The principal tools are the Covering Theorem for L and the technique of reverse Easton iteration.Let I denote the class of Silver indiscernibles for L and 〈iα ∣ α ϵ ORD〉 its increasing enumeration. Also fix an inner model M of GCH not containing 0# and let ωα denote the ωα of the model M[0#], the least inner model containing M as a submodel and 0# as an element.

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.

Universally Baire sets and definable well-orderings of the reals

2003 ◽  
Vol 68 (4) ◽  
pp. 1065-1081
Author(s):  
SY D. Friedman ◽  
Ralf Schindler
Keyword(s):  
Inner Models ◽  
Strong Cardinals

AbstractLet n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n − 2 strong cardinals) that every Σ1n-set of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses “David's trick” in the presence of inner models with strong cardinals.

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.

Sets constructible from sequences of ultrafilters

1974 ◽  
Vol 39 (1) ◽  
pp. 57-66 ◽  
Author(s):  
William J. Mitchell
Keyword(s):  
Continuum Hypothesis ◽  
Inner Models ◽  
Inner Model ◽  
Generalized Continuum

In [4], Kunen used iterated ultrapowers to show that ifUis a normalκ-complete nontrivial ultrafilter on a cardinalκthenL[U], the class of sets constructive fromU, has only the ultrafilterU∩L[U] and this ultrafilter depends only onκ. In this paper we extend Kunen's methods to arbitrary sequencesUof ultrafilters (provided that they fulfill a coherency condition) and obtain generalizations of these results (§3). In particular we answer Problem 1 of Kunen and Paris [5] which asks whether the number of ultrafilters onκcan be intermediate between 1 and 22κ. If there is a normalκ-complete ultrafilterUonκsuch that {α <κ: α is measurable} ∈Uthen there is an inner model with exactly two normal ultrafilters onκ, and ifκis super-compact then there are inner models havingκ+ +,κ+or any cardinal less than or equal toκnormal ultrafilters (§4).These methods also show that several properties ofLwhich had been shown to hold forL[U] also hold forL[U]: using an idea of Silver we show that inL[U] the generalized continuum hypothesis is true, there is a Souslin tree, and there is awell-ordering of the reals. In addition we generalize a result of Kunen to characterize the countaby complete ultrafilters ofL[U].

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.

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.

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