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Journal articles on the topic 'Singular Cardinal Hypothesis'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

NEEMAN, ITAY. "ARONSZAJN TREES AND FAILURE OF THE SINGULAR CARDINAL HYPOTHESIS." Journal of Mathematical Logic 09, no. 01 (June 2009): 139–57. http://dx.doi.org/10.1142/s021906130900080x.

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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.

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2

Matsubara, Yo. "Saturated ideals and the singular cardinal hypothesis." Journal of Symbolic Logic 57, no. 3 (September 1992): 970–74. http://dx.doi.org/10.2307/2275442.

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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.

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3

Mitchell, W. J. "On the singular cardinal hypothesis." Transactions of the American Mathematical Society 329, no. 2 (February 1, 1992): 507–30. http://dx.doi.org/10.1090/s0002-9947-1992-1073778-4.

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4

Viale, Matteo. "Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics." Bulletin of Symbolic Logic 14, no. 1 (March 2008): 99–113. http://dx.doi.org/10.2178/bsl/1208358846.

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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.

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5

Krueger, John. "Guessing models imply the singular cardinal hypothesis." Proceedings of the American Mathematical Society 147, no. 12 (August 7, 2019): 5427–34. http://dx.doi.org/10.1090/proc/14739.

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6

Friedman, Sy-David, and Radek Honzik. "A definable failure of the singular cardinal hypothesis." Israel Journal of Mathematics 192, no. 2 (March 28, 2012): 719–62. http://dx.doi.org/10.1007/s11856-012-0044-x.

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7

Poveda, Alejandro. "Contributions to the Theory of Large Cardinals through the Method of Forcing." Bulletin of Symbolic Logic 27, no. 2 (June 2021): 221–22. http://dx.doi.org/10.1017/bsl.2021.22.

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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.

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8

MAGIDOR, MENACHEM, and JOUKO VÄÄNÄNEN. "ON LÖWENHEIM–SKOLEM–TARSKI NUMBERS FOR EXTENSIONS OF FIRST ORDER LOGIC." Journal of Mathematical Logic 11, no. 01 (June 2011): 87–113. http://dx.doi.org/10.1142/s0219061311001018.

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We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim–Skolem–Tarski theorem for the equicardinality logic L(I), a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim–Skolem–Tarski theorem for the equicardinality logic at κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy.

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9

Apter, Arthur W., and Peter Koepke. "The consistency strength of choiceless failures of SCH." Journal of Symbolic Logic 75, no. 3 (September 2010): 1066–80. http://dx.doi.org/10.2178/jsl/1278682215.

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AbstractWe determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of ℵω. Using symmetric collapses to ℵω, , or , we show that injective failures at ℵω, , or can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell.

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10

Viale, Matteo. "The proper forcing axiom and the singular cardinal hypothesis." Journal of Symbolic Logic 71, no. 2 (June 2006): 473–79. http://dx.doi.org/10.2178/jsl/1146620153.

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11

Cummings, James, and Matthew Foreman. "Diagonal Prikry extensions." Journal of Symbolic Logic 75, no. 4 (December 2010): 1383–402. http://dx.doi.org/10.2178/jsl/1286198153.

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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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12

Gitik, Moti, and William J. Mitchell. "Indiscernible sequences for extenders, and the singular cardinal hypothesis." Annals of Pure and Applied Logic 82, no. 3 (December 1996): 273–316. http://dx.doi.org/10.1016/s0168-0072(96)00007-3.

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13

Abe, Yoshihiro. "Weakly normal ideals ou PKl and the singular cardinal hypothesis." Fundamenta Mathematicae 143, no. 2 (1993): 97–106. http://dx.doi.org/10.4064/fm-143-2-97-106.

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14

Gitik, Moti. "The strenght of the failure of the singular cardinal hypothesis." Annals of Pure and Applied Logic 51, no. 3 (March 1991): 215–40. http://dx.doi.org/10.1016/0168-0072(91)90016-f.

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15

Gitik, Moti. "The negation of the singular cardinal hypothesis from o(K)=K++." Annals of Pure and Applied Logic 43, no. 3 (August 1989): 209–34. http://dx.doi.org/10.1016/0168-0072(89)90069-9.

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16

Kojman, Menachem, and Saharon Shelah. "Nonexistence of universal orders in many cardinals." Journal of Symbolic Logic 57, no. 3 (September 1992): 875–91. http://dx.doi.org/10.2307/2275437.

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AbstractOur theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ1—a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

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17

MATSUBARA, YO, and SAHARON SHELAH. "NOWHERE PRECIPITOUSNESS OF THE NON-STATIONARY IDEAL OVER ${\mathcal P}_\kappa \lambda$." Journal of Mathematical Logic 02, no. 01 (May 2002): 81–89. http://dx.doi.org/10.1142/s021906130200014x.

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We prove that if λ is a strong limit singular cardinal and κ a regular uncountable cardinal < λ, then NSκλ, the non-stationary ideal over [Formula: see text], is nowhere precipitous. We also show that under the same hypothesis every stationary subset of [Formula: see text] can be partitioned into λκ disjoint stationary sets.

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18

Vasey, Sebastien. "Toward a stability theory of tame abstract elementary classes." Journal of Mathematical Logic 18, no. 02 (November 20, 2018): 1850009. http://dx.doi.org/10.1142/s0219061318500095.

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We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework.We also present an application to homogeneous model theory: for [Formula: see text] a homogeneous diagram in a first-order theory [Formula: see text], if [Formula: see text] is both stable in [Formula: see text] and categorical in [Formula: see text] then [Formula: see text] is stable in all [Formula: see text].

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19

Sinapova, Dima. "The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2." Journal of Symbolic Logic 77, no. 3 (September 2012): 934–46. http://dx.doi.org/10.2178/jsl/1344862168.

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20

BAGARIA, JOAN, and MENACHEM MAGIDOR. "ON ${\omega _1}$-STRONGLY COMPACT CARDINALS." Journal of Symbolic Logic 79, no. 01 (March 2014): 266–78. http://dx.doi.org/10.1017/jsl.2013.12.

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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 ${\kappa _0}$ .

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21

Zapletal, Jindřich. "Analytic Equivalence Relations and the Forcing Method." Bulletin of Symbolic Logic 19, no. 4 (September 2013): 473–90. http://dx.doi.org/10.1017/s107989860001057x.

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AbstractI describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.

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22

Sakai, Hiroshi, and Boban Veličković. "Stationary reflection principles and two cardinal tree properties." Journal of the Institute of Mathematics of Jussieu 14, no. 1 (November 1, 2013): 69–85. http://dx.doi.org/10.1017/s1474748013000315.

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AbstractWe study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.

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23

Moore, Justin Tatch. "Proper Forcing, Cardinal Arithmetic, and Uncountable Linear Orders." Bulletin of Symbolic Logic 11, no. 1 (March 2005): 51–60. http://dx.doi.org/10.2178/bsl/1107959499.

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AbstractIn this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of ℝ which is Σ1-definable in (H(ω2), ϵ). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, , C, C * where X is any suborder of the reals of size ω1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at k unless stationary subsets of reflect. The techniques are expected to be applicable to other open problems concerning the theory ofH(ω2).

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24

Steel, John R. "PFA implies ADL(ℝ)." Journal of Symbolic Logic 70, no. 4 (December 2005): 1255–96. http://dx.doi.org/10.2178/jsl/1129642125.

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In this paper we shall proveTheorem 0.1. Suppose there is a singular strong limit cardinal κ such that □κ fails; then AD holds in L(R).See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (αω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project.Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □κ fails for all uncountable cardinals κ. Thus we get immediately:It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(ℝ) and that PFA plus a measurable yields an inner model of ADℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models.In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply ADL(ℝ). (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of ADL(ℝ) from an “unaugmented” forcing axiom.

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25

CUMMINGS, JAMES, MATTHEW FOREMAN, and MENACHEM MAGIDOR. "SQUARES, SCALES AND STATIONARY REFLECTION." Journal of Mathematical Logic 01, no. 01 (May 2001): 35–98. http://dx.doi.org/10.1142/s021906130100003x.

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Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

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26

Gitik, Moti, and Menachem Magidor. "Extender based forcings." Journal of Symbolic Logic 59, no. 2 (June 1994): 445–60. http://dx.doi.org/10.2307/2275399.

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AbstractThe paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ > κ+ω” from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μ ≤ λ∣c f μ = ℵ0 and pp(μ) > λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ > κ+, and ”. In §4 a variation of the forcing of [The SCH revisited, §1] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying:“κ is a measurable and 2κ ≥ κ+α for some α, κ < c f α < α” starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.

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27

Cummings, James. "Moti Gitik. The strength of the failure of the singular cardinal hypothesis. Annals of pure and applied logic, vol. 51 (1991), pp. 215–240." Journal of Symbolic Logic 60, no. 1 (March 1995): 340. http://dx.doi.org/10.2307/2275527.

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28

Mitchell, W. J. "Moti Gitik. The negation of the singular cardinal hypothesis from O(K) = K++. Annals of pure and applied logic, vol. 43 (1989), pp. 209–234." Journal of Symbolic Logic 56, no. 1 (March 1991): 344. http://dx.doi.org/10.2307/2274937.

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29

Komjáth, Péter. "Some remarks on the partition calculus of ordinals." Journal of Symbolic Logic 64, no. 2 (June 1999): 436–42. http://dx.doi.org/10.2307/2586476.

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One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

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30

Gitik, Moti, and Peter Koepke. "Violating the singular cardinals hypothesis without large cardinals." Israel Journal of Mathematics 191, no. 2 (March 15, 2012): 901–22. http://dx.doi.org/10.1007/s11856-012-0028-x.

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31

Apter, Arthur W. "Successors of singular cardinals and measurability revisited." Journal of Symbolic Logic 55, no. 2 (June 1990): 492–501. http://dx.doi.org/10.2307/2274642.

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Before the remarkable theorem of Martin and Steel [6] showing that the existence of a supercompact cardinal κ implies L[R] ⊨ ZF + AD + DC, and the later theorem of Woodin [9] showing that Con(ZFC + There exists an ω sequence of Woodin cardinals) ⇔ Con(ZF + AD + DC), much set-theoretic research was focused upon showing that the consistency of fragments of AD + DC followed from more “reasonable” hypotheses such as versions of supercompactness. A good example of this is provided by the results of [1], in which the following theorems are proven.

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32

Schmerl, James H. "Some highly saturated models of Peano arithmetic." Journal of Symbolic Logic 67, no. 4 (December 2002): 1265–73. http://dx.doi.org/10.2178/jsl/1190150284.

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Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is a λ-saturated model of PA such that ∣M∣ > λ, then has an elementary extension of the same cardinality which is also λ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf() = λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

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33

Golshani, Mohammad. "The generalized Kurepa hypothesis at singular cardinals." Periodica Mathematica Hungarica 78, no. 2 (November 21, 2018): 200–202. http://dx.doi.org/10.1007/s10998-018-0267-7.

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Ismail, M., and A. Szymanski. "A topological equivalence of the singular cardinals hypothesis." Proceedings of the American Mathematical Society 123, no. 3 (March 1, 1995): 971. http://dx.doi.org/10.1090/s0002-9939-1995-1285997-0.

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Matsubara, Yo, and Toshimichi Usuba. "On skinny stationary subsets of." Journal of Symbolic Logic 78, no. 2 (June 2013): 667–80. http://dx.doi.org/10.2178/jsl.7802180.

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AbstractWe introduce the notion of skinniness for subsets of and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2λ-saturation of NSκλ ∣ X, where NSκλ denotes the non-stationary ideal over , implies the existence of a skinny stationary subset of X. We also show that if λ is a singular cardinal, then there is no skinnier stationary subset of . Furthermore, if λ is a strong limit singular cardinal, there is no skinny stationary subset of . Combining these results, we show that if λ is a strong limit singular cardinal, then NSκλ ∣ X can satisfy neither precipitousness nor 2λ-saturation for every stationary X ⊆ . We also indicate that , where , is equivalent to the existence of a skinnier (or skinniest) stationary subset of under some cardinal arithmetical hypotheses.

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Cummings, James, Yair Hayut, Menachem Magidor, Itay Neeman, Dima Sinapova, and Spencer Unger. "The ineffable tree property and failure of the singular cardinals hypothesis." Transactions of the American Mathematical Society 373, no. 8 (April 16, 2020): 5937–55. http://dx.doi.org/10.1090/tran/8110.

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Moore, Justin Tatch. "The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis." Annals of Pure and Applied Logic 140, no. 1-3 (July 2006): 128–32. http://dx.doi.org/10.1016/j.apal.2005.09.011.

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Cummings, James. "Moti Gitik and Menachem Magidor. The singular cardinal hypothesis revisited. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematical Sciences Research Institute publications, vol. 26, Springer-Verlag, New York etc. 1992, pp. 243–279." Journal of Symbolic Logic 60, no. 1 (March 1995): 339–40. http://dx.doi.org/10.2307/2275526.

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Kanamori, Akihiro. "Moti Gitik and Menachem Magidor. Extender based forcings. The Journal of Symbolic Logic, vol. 59 (1994), pp. 445–460. - Moti Gitik and William J. Mitchell. Indiscernible sequences for extenders, and the singular cardinal hypothesis. Annals of Pure and Applied Logic, vol. 82 (1996), pp. 273–316. - Moti Gitik. Blowing up the power of a singular cardinal. Annals of Pure and Applied Logic, vol. 80 (1996), pp. 17–33. - Moti Gitik and Carmi Merimovich. Possible values for and . Annals of Pure and Applied Logic, vol. 90 (1997), pp. 193–241. - Moti Gitik. Blowing up power of a singular cardinal—wider gaps. Annals of Pure and Applied Logic, vol. 116 (2002), pp. 1–38." Bulletin of Symbolic Logic 9, no. 2 (June 2003): 237–41. http://dx.doi.org/10.1017/s1079898600004637.

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Cummings, James. "Itay Neeman. Aronszajn trees and failure of the Singular Cardinal Hypothesis. Journal of Mathematical Logic, vol. 9, no. 1 (2009), pp. 139–157. - Dima Sinapova. The tree property at אּω+1. Journal of Symbolic Logic, vol. 77, no. 1 (2012), pp. 279–290. - Dima Sinapova. The tree property and the failure of SCH at uncountable cofinality. Archive for Mathematical Logic, vol. 51, no. 5-6 (2012), pp. 553–562. - Dima Sinapova. The tree property and the failure of the Singular Cardinal Hypothesis at אּω2. Journal of Symbolic Logic, vol. 77, no. 3 (2012), pp. 934–946. - Spencer Unger. Aronszajn trees and the successors of a singular cardinal. Archive for Mathematical Logic, vol. 52, no. 5-6 (2013), pp. 483–496. - Itay Neeman. The tree property up to אּω+1. Journal of Symbolic Logic. vol. 79, no. 2 (2014), pp. 429–459." Bulletin of Symbolic Logic 21, no. 2 (June 2015): 188–92. http://dx.doi.org/10.1017/bsl.2015.13.

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Matet, Pierre. "Yoshihiro Abe. Weakly normal filters and the closed unbounded filter on Pkλ. Proceedings of the American Mathematical Society, vol. 104 (1998), pp. 1226–1234. - Yoshihiro Abe. Weakly normal filters and large cardinals. Tsukuba journal of mathematics, vol. 16 (1992), pp. 487–494. - Yoshihiro Abe. Weakly normal ideals on Pkλ and the singular cardinal hypothesis. Fundamenta mathematicae, vol. 143 (1993), pp. 97–106. - Yoshihiro Abe. Saturation of fundamental ideals on Pkλ. Journal of the Mathematical Society of Japan, vol. 48 (1996), pp. 511–524. - Yoshihiro Abe. Strongly normal ideals on Pkλ and the Sup-function. opology and its applications, vol. 74 (1996), pp. 97–107. - Yoshihiro Abe. Combinatorics for small ideals on Pkλ. Mathematical logic quarterly, vol. 43 (1997), pp. 541–549. - Yoshihiro Abe and Masahiro Shioya. Regularity of ultrafilters and fixed points of elementary embeddings. Tsukuba journal of mathematics, vol. 22 (1998), pp. 31–37." Bulletin of Symbolic Logic 8, no. 2 (June 2002): 309–11. http://dx.doi.org/10.2178/bsl/1182353882.

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Mohammadpour, Rahman, and Boban Veličković. "Guessing models and the approachability ideal." Journal of Mathematical Logic, October 6, 2020, 2150003. http://dx.doi.org/10.1142/s0219061321500033.

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Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.

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Poveda, Alejandro, Assaf Rinot, and Dima Sinapova. "Sigma-Prikry forcing II: Iteration Scheme." Journal of Mathematical Logic, January 18, 2021, 2150019. http://dx.doi.org/10.1142/s0219061321500197.

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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.

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