On Jónsson cardinals with uncountable cofinality

1984 ◽  
Vol 49 (4) ◽  
pp. 315-324 ◽  
Author(s):  
Jan Tryba
Keyword(s):  
Jonsson 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.

DETERMINACY AND JÓNSSON CARDINALS INL(ℝ)

2014 ◽  
Vol 79 (4) ◽  
pp. 1184-1198 ◽  
Author(s):  
S. JACKSON ◽  
R. KETCHERSID ◽  
F. SCHLUTZENBERG ◽  
W. H. WOODIN
Keyword(s):  
Uncountable Cardinal ◽  
Jonsson Cardinals

AbstractAssume ZF + AD +V=L(ℝ) and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof (κ) = ω thenκis Rowbottom. We also establish some other partition properties.

scholarly journals Jónsson cardinals, Erdős cardinals, and the core model

1999 ◽  
Vol 64 (3) ◽  
pp. 1065-1086 ◽  
Author(s):  
W. J. Mitchell
Keyword(s):  
Minimal Model ◽  
Independent Interest ◽  
Core Model ◽  
Generic Extension ◽  
The Core ◽  
Steel Core ◽  
Inner Model ◽  
Jonsson Cardinals ◽  
Woodin Cardinal

AbstractWe show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.

On the Consistency Strength of `Accessible' Jonsson Cardinals and of the Weak Chang Conjecture.

1989 ◽  
Vol 54 (4) ◽  
pp. 1496
Author(s):  
Sy D. Friedman ◽  
Hans-Dieter Donder ◽  
Peter Koepke ◽  
Peter Koepke
Keyword(s):  
Consistency Strength ◽  
Jonsson Cardinals

On successors of Jónsson cardinals

2000 ◽  
Vol 39 (6) ◽  
pp. 465-473 ◽  
Author(s):  
J. Vickers ◽  
P.D. Welch
Keyword(s):  
Jonsson Cardinals

Jonsson-like partition relations and j: V → V

2004 ◽  
Vol 69 (4) ◽  
pp. 1267-1281 ◽  
Author(s):  
Arthur W. Apter ◽  
Grigor Sargsyan
Keyword(s):  
Axiom Of Choice ◽  
Elementary Embedding ◽  
Partition Relation ◽  
Partition Relations ◽  
Final Segment ◽  
The Axiom Of Choice ◽  
Jonsson Cardinals

Abstract.Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation . We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.

On the compactness of ℵ1 and ℵ2

1978 ◽  
Vol 43 (3) ◽  
pp. 394-401 ◽  
Author(s):  
C. A. di Prisco ◽  
J. Henle
Keyword(s):  
Recent Result ◽  
Large Cardinal ◽  
Partition Relations ◽  
Uncountable Cardinals ◽  
Open Question ◽  
Jonsson Cardinals

In recent years, the Axiom of Determinateness (AD) has yielded numerous results concerning the size and properties of the first ω-many uncountable cardinals. Briefly, these results began with Solovay's discovery that ℵ1 and ℵ2 are measurable [8], [3], continued with theorems of Solovay, Martin, and Kunen concerning infinite-exponent partition relations [6], [3], Martin's proof that ℵn has confinality ℵ2 for 1 < n < ω, and very recently, Kleinberg's proof that the ℵn are Jonsson cardinals [4].This paper was inspired by a very recent result of Martin from AD that ℵ1 is ℵ2-super compact. It was known for some time that AD implies ℵ1 is α-strongly compact for all ℵ < θ (where θ is the least cardinal onto which 2ω cannot be mapped, quite a large cardinal under AD), and that ADR implies that ℵ1, is α-super compact for all α < θ. A key open question had been whether or not ℵ1 is super compact under AD alone.This paper comments on the method of Martin in several different ways. In §2, we will prove that ℵ1 is ℵ2-super compact, and then generalize the method to show that ℵ2 is ℵ3-strongly compact. In addition, we will demonstrate a limitation in the method by showing that the possible measures obtained on are not normal, and that the method cannot be extended to show that ℵ2 is ℵ4-strongly compact.

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