scholarly journals Strong tree properties for small cardinals

2013 ◽  
Vol 78 (1) ◽  
pp. 317-333 ◽  
Author(s):  
Laura Fontanella
Keyword(s):  
Inaccessible Cardinal ◽  
Supercompact Cardinals

AbstractAn inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every n ≥ 2 and μ ≥ ℕn, we have (ℕn, μ)-ITP.

scholarly journals THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULAR CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 193-207 ◽  
Author(s):  
LAURA FONTANELLA
Keyword(s):  
Singular Limit ◽  
Inaccessible Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Image Position ◽  
Strongly Compact Cardinals

Abstract An inaccessible cardinal is strongly compact if, and only if, it satisfies the strong tree property. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that every successor of a singular limit of strongly compact cardinals has the strong tree property.

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

2015 ◽  
Vol 80 (1) ◽  
pp. 251-284
Author(s):  
SY-DAVID FRIEDMAN ◽  
PETER HOLY ◽  
PHILIPP LÜCKE
Keyword(s):  
Fixed Point ◽  
Measurable Cardinal ◽  
Large Cardinals ◽  
Inaccessible Cardinal ◽  
Proper Class ◽  
Supercompact Cardinals ◽  
Image Position ◽  
Class Forcing ◽  
Continuum Function ◽  
The Continuum

AbstractThis paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

On measurable limits of compact cardinals

1999 ◽  
Vol 64 (4) ◽  
pp. 1675-1688
Author(s):  
Arthur W. Apter
Keyword(s):  
Supercompact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Strongly Compact Cardinals

AbstractWe extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal κ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below κ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below κ with a non-trivial degree of supercompactness are the measurable limits of compact 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.

THE TREE PROPERTY UP TO אω+1

2014 ◽  
Vol 79 (2) ◽  
pp. 429-459 ◽  
Author(s):  
ITAY NEEMAN
Keyword(s):  
Large Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals

AbstractAssuming ω supercompact cardinals we force to obtain a model where the tree property holds both at אω+1, and at אn for all 2 ≤ n < ω. A model with the former was obtained by Magidor–Shelah from a large cardinal assumption above a huge cardinal, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our model, where the two hold simultaneously, is another step toward the goal of obtaining the tree property on increasingly large intervals of successor cardinals.

scholarly journals NONCONGLOMERABILITY FOR COUNTABLY ADDITIVE MEASURES THAT ARE NOT κ-ADDITIVE

2016 ◽  
Vol 10 (2) ◽  
pp. 284-300 ◽  
Author(s):  
MARK J. SCHERVISH ◽  
TEDDY SEIDENFELD ◽  
JOSEPH B. KADANE
Keyword(s):  
Conditional Probability ◽  
Inaccessible Cardinal ◽  
Conditional Probabilities ◽  
Uncountable Cardinal ◽  
Additive Probability ◽  
De Finetti ◽  
Additive Measures

AbstractLet κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

WEAK SQUARES AND VERY GOOD SCALES

2018 ◽  
Vol 83 (1) ◽  
pp. 1-12 ◽  
Author(s):  
MAXWELL LEVINE
Keyword(s):  
Supercompact Cardinal ◽  
Supercompact Cardinals ◽  
Good Scale ◽  
Weak Square

AbstractWe assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where □K,<K holds but □K, λ fails for λ < K.

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