Medium large cardinals

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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Large cardinals and gap-1 morasses

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2008.10.007 ◽

2009 ◽

Vol 159 (1-2) ◽

pp. 71-99 ◽

Cited By ~ 7

Author(s):

Andrew D. Brooke-Taylor ◽

Sy-David Friedman

Keyword(s):

Large Cardinals

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Co-stationarity of the ground model

Journal of Symbolic Logic ◽

10.2178/jsl/1154698589 ◽

2006 ◽

Vol 71 (3) ◽

pp. 1029-1043 ◽

Cited By ~ 3

Author(s):

Natasha Dobrinen ◽

Sy-David Friedman

Keyword(s):

Large Cardinals ◽

Partial Ordering ◽

Proper Class ◽

Ground Model ◽

Covering Theorem ◽

Uncountable Cardinal ◽

Cohen Forcing ◽

Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

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Some results on consecutive large cardinals

Annals of Pure and Applied Logic ◽

10.1016/0168-0072(83)90051-9 ◽

1983 ◽

Vol 25 (1) ◽

pp. 1-17 ◽

Cited By ~ 11

Author(s):

Arthur W. Apter

Keyword(s):

Large Cardinals

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Large Cardinals and Topology: a Short Retrospective and Some New Results

Logic Journal of IGPL ◽

10.1093/jigpal/jzm052 ◽

2007 ◽

Vol 15 (5-6) ◽

pp. 433-443 ◽

Cited By ~ 2

Author(s):

S. G. Da Silva

Keyword(s):

Large Cardinals ◽

And Topology

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Homogeneous changes in cofinalities with applications to HOD

Journal of Mathematical Logic ◽

10.1142/s0219061317500076 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750007 ◽

Cited By ~ 1

Author(s):

Omer Ben-Neria ◽

Spencer Unger

Keyword(s):

Large Cardinals ◽

An AD-like model

Journal of Symbolic Logic ◽

10.2307/2274240 ◽

1985 ◽

Vol 50 (2) ◽

pp. 531-543 ◽

Cited By ~ 3

Author(s):

Arthur W. Apter

Keyword(s):

Measurable Cardinal ◽

Large Cardinals ◽

Strong Sense ◽

Supercompact Cardinals ◽

Singular Cardinals ◽

Partial Orderings ◽

Measurable Cardinals ◽

Jonsson Cardinals

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].Another such model will be constructed in this paper. Specifically, the following theorem is proven.Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.

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

Bulletin of Symbolic Logic ◽

10.1017/bsl.2021.22 ◽

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.

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