Sigma-Prikry forcing II: Iteration Scheme

Poveda, Alejandro; Rinot, Assaf; Sinapova, Dima

doi:10.1142/s0219061321500197

Cited by 9 publications

(17 citation statements)

References 4 publications

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“…Similarly, the same wish extends to singular cardinals, although in this context one will encounter a shortage of parallels of PFA. A potential strategy to overcome this problem might bear on the abstract iteration scheme for singular cardinals introduced in the Σ-Prikry project by Rinot, Sinapova and the third author [PRS21b,PRS21a,PRS21c].…”

Section: Open Problemsmentioning

confidence: 99%

Negating the Galvin Property

Tom¹,

Garti²,

Poveda³

2021

Preprint

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We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property, and explores stronger forms of this property in the context of large cardinals. Contents 0. Introduction 1. Preliminaries 1.1. Notations 1.2. Clubs and Filters 1.3. Forcing preliminaries 1.4. Radin forcing with interleaved collapses 2. The consistency of the local and global failure 2.1. Local failure of Galvin's property 2.2. Global failure of Galvin's property 2.3. The impossibility of the ultimate global failure 2.4. Galvin's number at successor of singular cardinals 3. The strong failure and Prikry-type generic extensions 4. Stronger forms of Galvin's property on normal filters 5. Open problems 5.1. The failure at successors of singulars 5.2. Galvin's property and large cardinals 5.3. Consistency strength of the failure of Galvin's property 2010 Mathematics Subject Classification. 03E35, 03E55.

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Section: Open Problemsmentioning

confidence: 99%

Negating the Galvin Property

Tom¹,

Garti²,

Poveda³

2021

Preprint

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“…This forcing plays an important role in the proof of Theorem 1.2. In a sequel to this paper [18], we will describe this forcing in detail and prove that it is Σ-Prikry, where Σ ∶= ⟨κ n | n < ω⟩.…”

Section: Extender-based Prikry Forcingmentioning

confidence: 99%

“…The exact definition of forking projection may be found in Section 4, but, roughly speaking, this is a kind of projection that ensures a much better correspondence between the two Σ-Prikry triples, which later allows to iterate this procedure. In a sequel to this paper [18], we present our iteration scheme for Σ-Prikry notions of forcing, from which we obtain a correct proof of (a strong form of) Sharon's result: Theorem 1.2 Suppose that ⟨κ n | n < ω⟩ is a strictly increasing sequence of Laverindestructible supercompact cardinals. Denote κ ∶= sup n<ω κ n .…”

Section: Introductionmentioning

confidence: 99%

“…19 For this, in the next section we shall devise a notion of forcing for killing a given single counterexample to Refl(1, E μ ω , ). Then, in [18], we find a mean to iterate it.…”

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“…Note that, as Pn is κn-closed, (Eμ <κn ) V Pn = (E μ <κn ) V 18. ⟨T i | i < k⟩ stands for the G-interpretation of the sequence of P-names ⟨τ i | i < k⟩.…”

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Sigma-Prikry forcing I: The Axioms

Poveda¹,

Rinot

Sinapova³

2020

Can. J. Math.-J. Can. Math.

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We introduce a class of notions of forcing which we call $\Sigma $ -Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma $ -Prikry. We show that given a $\Sigma $ -Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set T, there exists a corresponding $\Sigma $ -Prikry poset that projects to $\mathbb P$ and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma $ -Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $\kappa $ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa $ remains a strong limit cardinal, every finite collection of stationary subsets of $\kappa ^+$ reflects simultaneously, and $2^\kappa =\kappa ^{++}$ .

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