A model for a very good scale and a bad scale

Sinapova, Dima

doi:10.2178/jsl/1230396925

Cited by 15 publications

(33 citation statements)

References 6 publications

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“…The proof of this claim adapts an argument due to Solovay, Reinhardt, and Kanamori [7]. For details, see Lemma 2 in [6]. Now, define the chain as follows.…”

Section: The Constructionmentioning

confidence: 83%

“…In 2008 Gitik-Sharon [2] showed two important consistency results about scales: that failure of SCH does not imply weak square, and the existence of a very good scale does not imply weak square. The result was generalized by Sinapova [6] for singular cardinals of arbitrary cofinality. The Gitik-Sharon model provided much of the motivation behind the construction in Neeman [5].…”

Section: Theoremmentioning

confidence: 95%

“…In section 2 we define the forcing notion and give some basic properties about the forcing. The forcing that we will use combines ideas from Neeman [5] and Sinapova [6]. Also, we describe the very good scale and the bad scale in the generic extension.…”

Section: Theoremmentioning

confidence: 99%

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The tree property and the failure of SCH at uncountable cofinality

Sinapova

2012

Arch. Math. Logic

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“…The proof of this claim adapts an argument due to Solovay, Reinhardt, and Kanamori [7]. For details, see Lemma 2 in [6]. Now, define the chain as follows.…”

Section: The Constructionmentioning

confidence: 83%

Section: Theoremmentioning

confidence: 95%

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The tree property and the failure of SCH at uncountable cofinality

Sinapova

2012

Arch. Math. Logic

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“…In her doctoral thesis Dima Sinapova answered this question. Sinapova [27] introduced a version of the Gitik-Sharon forcing which makes a supercompact cardinal κ into a singular cardinal of uncountable cofinality, and showed that in her model there are cofinal sets carrying a non-good scale and a very good scale. (2) Itay Neeman [23] used a variant of the Gitik-Sharon construction to produce a model in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of cofinality ω, and κ + has the tree property.…”

Section: Proof We Work In the Modelmentioning

confidence: 99%

Diagonal Prikry extensions

Cummings¹,

Foreman

2010

J. symb. log.

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§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

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“…The proof adapts easily to produce other failures 2 κ = λ > κ + , but one has to increase the large cardinal assumption to δ supercompactness for δ > (ν + ) V . The work of Gitik-Sharon and Cummings-Foreman was generalized by Sinapova [26] to produce an extension with arbitrary cofinality for κ. It is likely, but not known, that similar generalizations are possible with our construction.…”

mentioning

confidence: 99%

Aronszajn Trees and Failure of the Singular Cardinal Hypothesis

Neeman

2009

J. Math. Log.

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Abstract. The tree property at κ + states that there are no Aronszajn trees on κ + , or, equivalently, that every κ + tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ + and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper we reconcile the two. We prove from large cardinals that the tree property at κ + is consistent with failure of the singular cardinal hypothesis at κ. §1. Introduction. In the early 1980s Woodin asked whether failure of the singular cardinal hypothesis (SCH) at ℵ ω implies the existence of an Aronszajn tree on ℵ ω+1 . More generally, in 1989 Woodin and others asked whether failure of the SCH at a cardinal κ of cofinality ω, implies the existence of an Aronszajn tree on κ + , see Foreman [7, §2]. To understand the motivation for the question let us recall some results surrounding the SCH and trees in infinitary combinatorics.The singular cardinal hypothesis, in its most specific form, states that 2 κ = κ + whenever κ is a singular strong limit cardinal. (There are several forms that are more general. For example the statement that κ cof(κ) = κ + whenever κ is singular and 2 cof(κ) < κ. Or the statement that for every singular cardinal κ, 2 κ is as small as it can be, subject to two requirements: monotonicity, namely that 2 κ ≥ sup{2 δ | δ < κ}; and König's theorem, which implies cof(2 κ ) > κ. Both these forms imply the specific form, that 2 κ = κ + whenever κ is a singular strong limit cardinal.)Cohen forcing of course shows that the parallel hypothesis for regular cardinals is consistently false. Indeed it can be made to fail in any arbitrary way, subject to monotonicity and König's theorem. For a while after the introduction of forcing it was expected that the same should hold for the SCH, and that proving this was only a matter of discovering sufficiently sophisticated forcing notions. Some progress was made in this direction, and ultimately led to models with failure of the SCH described below. But it turned out that changing the power of a singular cardinal is much harder than changing the power of a regular cardinal, and in some cases it is outright impossible. The first indication of this was a theorem of Silver [25], that the continuum hypothesis cannot fail for the first time at a singular cardinal of uncountable cofinality. Another is a theorem of Solovay [27], that the SCH holds above a strongly compact. The most celebrated is a theorem of Shelah [23], that if 2 ℵn < ℵ ω for each n < ω, then 2 ℵω < ℵ ω4 . Still, the SCH can be made to fail. One way to violate the hypothesis is to start with a measurable cardinal κ, make the continuum hypothesis fail at κ, for example by increasing the power set of κ to κ ++ -an easy task as κ is regular, not singular-and then, assuming κ remained measurable, use Prikry forcing to turn its cofinality to ω. In the resulti...

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A model for a very good scale and a bad scale

Cited by 15 publications

References 6 publications

The tree property and the failure of SCH at uncountable cofinality

The tree property and the failure of SCH at uncountable cofinality

Diagonal Prikry extensions

Aronszajn Trees and Failure of the Singular Cardinal Hypothesis

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