On the existence of universal models

Džamonja, Mirna; Shelah, Saharon

doi:10.1007/s00153-004-0235-1

Cited by 17 publications

(28 citation statements)

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“…When κ is singular the strategy is no longer available, and this is one difficulty among many in proving consistency results involving singular cardinals and their successors. Džamonja and Shelah [7] introduced a new idea, which we briefly describe. Initially κ is a supercompact cardinal whose supercompactness is indestructible under <κ-directed closed forcing.…”

Section: Introductionmentioning

confidence: 99%

A framework for forcing constructions at successors of singular cardinals

Cummings

Džamonja

Magidor

et al. 2017

Trans. Amer. Math. Soc.

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Abstract. We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable cofinality, while κ + enjoys various combinatorial properties.As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable cofinality where SCH fails and such that there is a collection of size less than 2 κ + of graphs on κ + such that any graph on κ + embeds into one of the graphs in the collection.2010 Mathematics Subject Classification. Primary: 03E35, 03E55, 03E75.

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Section: Introductionmentioning

confidence: 99%

A framework for forcing constructions at successors of singular cardinals

Cummings

Džamonja

Magidor

et al. 2017

Trans. Amer. Math. Soc.

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“…⋆ 2.2 Things change at the successor of a singular! Positive results analogous to the Džamonja-Shelah [4] were obtained for κ of countable cofinality by Džamonja and Shelah in [3] and for arbitrary cofinality by Cummings, Džamonja, Magidor, Morgan and Shelah in [2]. We quote that general result: Theorem 2.3 (Cummings et al [2]) If κ is a supercompact cardinal, λ < κ is a regular cardinal and Θ is a cardinal with cf(Θ) ≥ κ ++ and κ +3 ≤ Θ there is a forcing extension in which cf(κ) = λ, 2 κ = 2 κ + = Θ and there is a universal family of graphs on κ + of size κ ++ .…”

Section: Universal Graphsmentioning

confidence: 82%

The Singular World of Singular Cardinals

Džamonja¹

2015

Logic Without Borders

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The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones. 1 IntroductionInfinite cardinals can be regular or singular. Regular cardinals and especially successors of regular cardinals, tend to lend themselves to easier and better understood combinatorial methods and hence are often considered as being in some sense easier. For example, Todd Eisworth in his Handbook of Set Theory article [6] artfully exposes difficulties one has in dealing with the combinatorics of the successors of singulars and explains on a number of examples why the methods used at a successor of a regular most often cannot work when dealing with the successor of a singular. Indeed, it is known that in many situations, dealing with singular cardinals and their successors has to involve techniques beyond combinatorics and forcing, and it notably requires large cardinals. This is true even for such seemingly elementary properties as the calculation of the size of the power set of the cardinal κ as a function of the size of the power sets of the cardinals below, which is basically the content of the famous Singular Cardinal Hypothesis and which has lead to some of the deepest results throughout set theory. In fact, the common wisdom and the thesis of [6] are that if the universe is close to L then the singular cardinals and their successors are "manageable", and the opposite is true in the models obtained by using strong enough large cardinal hypothesis.We shall explore the antithesis, which is that (a) in some situations singular cardinals are more manageable than the regular ones and (b) in some models obtained from large 2010 Mathematics subject classification. 03E05

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“…(1) It is possible to produce models where u κ + is arbitrarily large [6], for example by adding many Cohen subsets of κ over a model of GCH. (2) It is possible to produce models where κ <κ = κ, 2 κ is arbitrarily large and u κ + = κ ++ [4] by iterated forcing over a model of GCH.…”

Section: Introductionmentioning

confidence: 99%

“…Our final model will be obtained by halting the construction at a suitable stage i * of cofinality κ ++ , and forcing with P i * . The point here (an idea which comes from [4]) is that we can read off a universal family of size κ ++ from a cofinal set of stages below i * , and we are in a situation where 2 κ = κ +3 .…”

Section: Introductionmentioning

confidence: 99%