SET FORCING AND STRONG CONDENSATION FORH(ω2)

2015 ◽  
Vol 80 (1) ◽  
pp. 56-84
Author(s):  
LIUZHEN WU
Keyword(s):  
Large Cardinal ◽  
Partial Answer ◽  
Supercompact Cardinal ◽  
Abstract Version ◽  
Strong Condensation

AbstractThe Axiom of Strong Condensation, first introduced by Woodin in [14], is an abstract version of the Condensation Lemma ofL. In this paper, we construct a set-sized forcing to obtain Strong Condensation forH(ω2). As an application, we show that “ZFC + Axiom of Strong Condensation +”is consistent, which answers a question in [14]. As another application, we give a partial answer to a question of Jech by proving that “ZFC + there is a supercompact cardinal + any ideal onω1which is definable overH(ω2) is not precipitous” is consistent under sufficient large cardinal assumptions.

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.

SUITABLE EXTENDER MODELS I

2010 ◽  
Vol 10 (01n02) ◽  
pp. 101-339 ◽  
Author(s):  
W. HUGH WOODIN
Keyword(s):  
Strong Evidence ◽  
Large Cardinal ◽  
Central Issue ◽  
Supercompact Cardinal ◽  
Model Program ◽  
Inner Model

We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the central issue for the Inner Model Program.

scholarly journals Indestructibility and the level-by-level agreement between strong compactness and supercompactness

2002 ◽  
Vol 67 (2) ◽  
pp. 820-840 ◽  
Author(s):  
Arthur W. Apter ◽  
Joel David Hamkins
Keyword(s):  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Strong Compactness

AbstractCan a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets. or less level-by-level agreement, such as requiring it only on measure one sets, then yes. it can.

MRP, tree properties and square principles

2011 ◽  
Vol 76 (4) ◽  
pp. 1441-1452 ◽  
Author(s):  
Remi Strullu
Keyword(s):  
Direct Proof ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Square Principles ◽  
Proper Forcing ◽  
The Relationship

AbstractWe show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

Stationary reflections for uncountable cofinality

1986 ◽  
Vol 51 (1) ◽  
pp. 147-151 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Partial Order ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Stationary Reflection ◽  
Supercompact Cardinals ◽  
Stationary Set ◽  
Compactness Theorems ◽  
Weakly Compact Cardinal

It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is Lévy-collapsed to ω2 the new ω2 inherits some of the large cardinal properties; e.g. if S is a stationary set of ω-limits in ω2 then for some α < ω2, S ∩ α is stationary in α. Later S. Shelah extended this to the following theorem: if a supercompact cardinal κ is Lévy-collapsed to ω2, then in the resulting model the following holds: if S ⊆ λ is a stationary set of ω-limits and cf(λ) ≥ ω2 then there is an α. < λ such that S ∩ α is stationary in α, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality μ as well, i.e. when an appropriate large cardinal is collapsed to μ++. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of ω-limits remains stationary after forcing with an ω2-closed partial order may be false in the case of μ-limits in a cardinal of the form λ+ with cf(λ) < μ, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality ≤ μ exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond μ.

On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ

1993 ◽  
Vol 58 (1) ◽  
pp. 119-127
Author(s):  
Julius B. Barbanel
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
The Other ◽  
Supercompact Cardinal ◽  
Partition Property ◽  
Other Hand ◽  
The One ◽  
Measurable Cardinals ◽  
The Relationship ◽  
Normal Ultrafilter

AbstractSuppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.

Supercompact cardinals, trees of normal ultrafilters, and the partition property

1986 ◽  
Vol 51 (3) ◽  
pp. 701-708
Author(s):  
Julius B. Barbanel
Keyword(s):  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Supercompact Cardinals ◽  
Partition Property

AbstractSuppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on Pκ(λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.

Triharmonic Riemannian submersions from 3-dimensional space forms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tomoya Miura ◽  
Shun Maeta
Keyword(s):  
Existence Theorem ◽  
Space Form ◽  
Dimensional Space ◽  
Riemannian Submersion ◽  
Partial Answer ◽  
Riemannian Submersions ◽  
Space Forms ◽  
3 Dimensional

Abstract We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.

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