Saturated ideals and the singular cardinal hypothesis

1992 ◽  
Vol 57 (3) ◽  
pp. 970-974 ◽  
Author(s):  
Yo Matsubara
Keyword(s):  
Large Cardinal ◽  
Inaccessible Cardinal ◽  
Elementary Embedding ◽  
Normal Ideal ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Singular Cardinal Hypothesis ◽  
Generic Filter ◽  
Generic Ultrapowers

The large cardinal-like properties of saturated ideals have been investigated by various authors, including Foreman [F], and Jech and Prikry [JP], among others. One of the most interesting consequences of a strongly compact cardinal is the following theorem of Solovay [So2]: if a strongly compact cardinal exists then the singular cardinal hypothesis holds above it. In this paper we discuss the question of relating the existence of saturated ideals and the singular cardinal hypothesis. We will show that the existence of “strongly” saturated ideals implies the singular cardinal hypothesis. As a biproduct we will present a proof of the above mentioned theorem of Solovay using generic ultrapowers. See Jech and Prikry [JP] for a nice exposition of generic ultrapowers. We owe a lot to the work of Foreman [F]. We would like to express our gratitude to Noa Goldring for many helpful comments and discussions.Throughout this paper we assume that κ is a strongly inaccessible cardinal and λ is a cardinal >κ. By an ideal on κλ we mean a κ-complete fine ideal on Pκλ. For I an ideal on κλ let PI denote the poset of I-positive subsets of κλ.Definition. Let I be an ideal on κλ. We say that I is a bounding ideal if 1 ⊩-PI “δ(δ is regular cardinal ”.We can show that if a normal ideal is “strongly” saturated then it is bounding.Theorem 1. If 1 is an η-saturated normal ideal onκλ, where η is a cardinal <λsuch that there are fewer thanκmany cardinals betweenκand η (i.e. η < κ+κ), then I is bounding.Proof. Let I be such an ideal on κλ. By the work of Foreman [F] and others, we know that every λ+-saturated normal ideal is precipitous. Suppose G is a generic filter for our PI. Let j: V → M be the corresponding generic elementary embedding. By a theorem of Foreman [F, Lemma 10], we know that Mλ ⊂ M in V[G]. By η-saturation, cofinalities ≥η are preserved; that is, if cfvα ≥ η, then cfvα = cfv[G]α. From j ↾ Vκ being the identity on Vκ and M being λ-closed in V[G], we conclude that cofinalities <κ are preserved. Therefore if cfvα ≠ cfv[G]α then κ ≤ cfvα < η.

On the ordering of certain large cardinals

1979 ◽  
Vol 44 (4) ◽  
pp. 563-565
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Inaccessible Cardinal ◽  
Elementary Embedding ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
The Universe ◽  
Weakly Compact Cardinal

It is well known that the first strongly inaccessible cardinal is strictly less than the first weakly compact cardinal which in turn is strictly less than the first Ramsey cardinal, etc. However, once one passes the first measurable cardinal the inequalities are no longer strict. Magidor [3] has shown that the first strongly compact cardinal may be equal to the first measurable cardinal or equal to the first super-compact cardinal (the first supercompact cardinal is strictly larger than the first measurable cardinal). In this note we will indicate how Magidor's methods can be used to show that it is undecidable whether one cardinal (the first strongly compact) is greater than or less than another large cardinal (the first huge cardinal). We assume that the reader is familiar with the ultrapower construction of Scott, as presented in Drake [1] or Kanamori, Reinhardt and Solovay [2].Definition. A cardinal κ is huge (or 1-huge) if there is an elementary embedding j of the universe V into a transitive class M such that M contains the ordinals, is closed under j(κ) sequences, j(κ) > κ and j ↾ Rκ = id. Let κ denote the first huge cardinal, and let λ = j(κ).One can see from easy reflection arguments that κ and λ are inaccessible in V and, in fact, that κ is measurable in V.

scholarly journals TAMENESS FROM LARGE CARDINAL AXIOMS

2014 ◽  
Vol 79 (4) ◽  
pp. 1092-1119 ◽  
Author(s):  
WILL BONEY
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Dual Property ◽  
First Time ◽  
Strongly Compact Cardinals ◽  
Do So

AbstractWe show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS(K) below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, calledtype shortness, and show that it follows similarly from large cardinals.

scholarly journals WOODIN FOR STRONG COMPACTNESS CARDINALS

2019 ◽  
Vol 84 (1) ◽  
pp. 301-319
Author(s):  
STAMATIS DIMOPOULOS
Keyword(s):  
Identity Crisis ◽  
Large Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Strong Compactness ◽  
Definition Of ◽  
Vopěnka Cardinals

AbstractWoodin and Vopěnka cardinals are established notions in the large cardinal hierarchy and it is known that Vopěnka cardinals are the Woodin analogue for supercompactness. Here we give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor’s identity crisis theorem for the first strongly compact cardinal.

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

2008 ◽  
Vol 14 (1) ◽  
pp. 99-113
Author(s):  
Matteo Viale
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Forcing Axioms ◽  
Supercompact Cardinals ◽  
Saturation Properties ◽  
Proper Forcing Axiom ◽  
Singular Cardinals ◽  
Proper Forcing

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.

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.

Models with second order properties in successors of singulars

1989 ◽  
Vol 54 (1) ◽  
pp. 122-137
Author(s):  
Rami Grossberg
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
First Order ◽  
First Order Theory

AbstractLet L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p.Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ<λ. It is new when λ is singular or when ∣T∣ = λ is regular.Theorem 3. Let λ be singular. If Con(ZFC + GCH + ∃κ) [κ is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false.

ON ${\omega _1}$-STRONGLY COMPACT CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 266-278 ◽  
Author(s):  
JOAN BAGARIA ◽  
MENACHEM MAGIDOR
Keyword(s):  
Measurable Cardinal ◽  
Second Order ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Relational Structures ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Strongly Compact Cardinals ◽  
Order Reflection

Abstract An uncountable cardinal κ is called ${\omega _1}$ -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$ -complete ultrafilter on I. We show that the first ${\omega _1}$ -strongly compact cardinal, ${\kappa _0}$ , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$ . We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$ . Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$ .

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

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.

Ramsey-like cardinals

2011 ◽  
Vol 76 (2) ◽  
pp. 519-540 ◽  
Author(s):  
Victoria Gitman
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Large Fragment ◽  
Weakly Compact ◽  
Elementary Embeddings ◽  
Measurable Cardinals

AbstractOne of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V = L.

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