Hybrid Prikry forcing

2015 ◽  
Vol 228 (2) ◽  
pp. 139-152
Author(s):  
Dima Sinapova
Keyword(s):  
Prikry Forcing

scholarly journals Contributions to the Theory of Large Cardinals through the Method of Forcing

2021 ◽  
Vol 27 (2) ◽  
pp. 221-222
Author(s):  
Alejandro Poveda
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Prikry Forcing ◽  
Regular Cardinals ◽  
Singular Cardinals ◽  
Vopěnka’S Principle

AbstractThe dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle (Part I). In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics (Part II and Part III).We commence Part I by investigating the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{(n)}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $(\lambda ^{+\omega })^{\mathrm {HOD}}<\lambda ^+$ , for every regular cardinal $\lambda $ .” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $(\lambda ^+)^{\textrm {{HOD}}}=\lambda ^+$ , there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$ , whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation (Part III) we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously.

scholarly journals The subcompleteness of diagonal Prikry forcing

2019 ◽  
Vol 59 (1-2) ◽  
pp. 81-102
Author(s):  
Kaethe Minden
Keyword(s):  
Prikry Forcing

Partition properties and Prikry forcing on simple spaces

1990 ◽  
Vol 55 (3) ◽  
pp. 938-947
Author(s):  
J. M. Henle
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Short Introduction ◽  
Prikry Forcing ◽  
Partition Property ◽  
Combinatorial Objects ◽  
The Relationship

One of the simplest and yet most fruitful ideas in forcing was the notion of Karel Prikry in which he used a measure on a cardinal κ to change the cofinality of κ to ω without collapsing it. The idea has found connections to almost every branch of modern set theory, from large cardinals to small, from combinatorics to models, from Choice to Determinacy, and from consistency to inconsistency. The long list of generalizers and modifiers includes Apter, Gitik, Henle, Spector, Shelah, Mathias, Magidor, Radin, Blass and Kimchi.This paper is about generalizing Prikry forcing and partition properties to “simple spaces”. The concept of a simple space is itself the generalization of those combinatorial objects upon which the notions of “measurable”, “compact”, “supercompact”, “huge”, etc. are based. Simple spaces were introduced in [ADHZ1] and [ADHZ2] together with a broader generalization, “filter spaces”. The definition provided here is a small simplification of earlier versions. The author is indebted to Mitchell Spector, whose careful reading turned up numerous errors, some subtle, some flagrant.In this first section, we review simple spaces briefly, including a short introduction to the space Qκλ. In §2, we describe our generalizations of partition property and Prikry forcing, and discuss the relationship between them. In §3, we find a partition property for the huge space [λ]κ, but show that Prikry forcing here is impossible. We find partition properties for Qκλ and show that Prikry forcing can be done here.

scholarly journals SMALL UNIVERSAL FAMILIES OF GRAPHS ON ℵω + 1

2016 ◽  
Vol 81 (2) ◽  
pp. 541-569 ◽  
Author(s):  
JAMES CUMMINGS ◽  
MIRNA DŽAMONJA ◽  
CHARLES MORGAN
Keyword(s):  
Strong Limit ◽  
Prikry Forcing ◽  
Image Position

AbstractWe prove that it is consistent that $\aleph _\omega $ is strong limit, $2^{\aleph _\omega } $ is large and the universality number for graphs on $\aleph _{\omega + 1} $ is small. The proof uses Prikry forcing with interleaved collapsing.

scholarly journals Sigma-Prikry forcing II: Iteration Scheme

2021 ◽  
pp. 2150019
Author(s):  
Alejandro Poveda ◽  
Assaf Rinot ◽  
Dima Sinapova
Keyword(s):  
Blow Up ◽  
General Scheme ◽  
Iteration Scheme ◽  
Singular Cardinal ◽  
Prikry Forcing ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis ◽  
Singular Cardinals ◽  
Stationary Set ◽  
Stationary Subsets

In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

Prikry forcing at κ+ and beyond

1987 ◽  
Vol 52 (1) ◽  
pp. 44-50
Author(s):  
William Mitchell
Keyword(s):  
Specific Measure ◽  
Prikry Forcing ◽  
Normal Measure

If U is a normal measure on κ then we can add indiscernibles for U either by Prikry forcing [P] or by taking an iterated ultrapower which will add a sequence of indiscernibles for over M. These constructions are equivalent: the set C of indiscernibles for added by the iterated ultrapower is Prikry generic for [Mat]. Prikry forcing has been extended for sequences of measures of length by Magidor [Mag], and his method readily extends to . In this case the measure U is replaced by a sequence of measures and the set C of indiscernibles is replaced by a system of indiscernibles for : is a function such that (κ, β) is a set of indiscernibles for (κ, β) for each . The equivalence between forcing and iterated ultra-powers still holds true for such sequences: there is an interated ultrapower j: V → M (which is defined in detail later in this paper) such that the system of indiscernibles for j() constructed by j is Magidor generic over M.The construction of the system of indiscernibles works equally well for o(κ) ≧ κ+. Radin has defined a variant of Prikry forcing which also works for o(κ) > κ+ ([R]; see also [Mi82] where Radin forcing is applied specifically to sequences of measures, rather than to hypermeasures as in Radin's paper), but Radin's forcing is weaker than Magidor's extension of Prikry forcing in the sense that the system of indiscernibles generated by the interated ultrapower is not Radin generic for j(), but only the set . That is, an indiscernible does not belong to a specific measure, but only to the whole sequence of measures on the cardinal κ.

Supercompact extender based Prikry forcing

2011 ◽  
Vol 50 (5-6) ◽  
pp. 591-602 ◽  
Author(s):  
Carmi Merimovich
Keyword(s):  
Prikry Forcing
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