Large Cardinal Properties Research Articles

Indestructibility properties of Ramsey and Ramsey-like cardinals

We develop new techniques, for use in indestructibility arguments, of lifting embeddings on transitive set models of ZFC− which lack closure and of lifting iterations of such embeddings to a forcing extension. We use these techniques to establish basic indestructibility results for Ramsey and the Ramsey-like cardinals – α-iterable, strongly Ramsey, and super Ramsey cardinals – introduced in [11]. We show that Ramsey, α-iterable, strongly Ramsey, and super Ramsey cardinals κ are indestructible by small forcing, the canonical forcing of the GCH, and the forcing to add a fast function on κ. We also show that if κ is one of these large cardinals, then there is a forcing extension in which the large cardinal property of κ becomes indestructible by Add(κ,θ) for any cardinal θ.The following are consequences of the indestructibility results. If κ is Ramsey, α-iterable, strongly Ramsey, or super Ramsey, then there is a forcing extension preserving this in which the GCH fails at κ. If κ is one of these large cardinals, then there is a forcing extension preserving the large cardinal property of κ in which κ is not even weakly compact in HOD. If κ is Ramsey, then there is a forcing extension in which κ remains virtually Ramsey, but is no longer Ramsey (this answers positively a question posed in [11]).

Structural reflection, shrewd cardinals and the size of the continuum

Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [Formula: see text] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [Formula: see text] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [Formula: see text]-indescribability, can be canonically characterized through localized versions of the principle [Formula: see text]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [Formula: see text]-indescribability.

More on HOD-supercompactness

We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a proper class of supercompact cardinals. Finally, we also produce a model in which the unique supercompact cardinal is also the only strongly compact cardinal, no cardinal is supercompact up to an inaccessible cardinal, level by level inequivalence holds and the unique supercompact cardinal is not HOD-supercompact.

Small models, large cardinals, and induced ideals

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.

A REFINEMENT OF THE RAMSEY HIERARCHY VIA INDESCRIBABILITY

AbstractWe study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $ . By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s $\Pi _\alpha $ -Ramsey and $\Pi _{\alpha +1}$ -Ramsey cardinals for all odd $\alpha <\omega $ and for all $\omega \leq \alpha <\kappa $ . We also show that, given any ordinals $\beta _0,\beta _1<\kappa $ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi ^1_{\beta _0}$ -indescribability ideal and the $\Pi ^1_{\beta _1}$ -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi ^1_\xi $ -indescribability and Ramseyness.

Partition properties for simply definable colourings

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of ω1 for every colouring of the finite sets of countable ordinals that is definable by a Σ1-formula that only uses the cardinal ω1 and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given large cardinal and these implications yield natural examples of inaccessible cardinals that possess strong partition properties for Σ1-definable colourings and are not weakly compact. In contrast, we show that Σ1- definability behaves fundamentally different at ω2 by showing that various large cardinal assumptions and Martin’s Maximum are compatible with the existence of a colouring of pairs of elements of ω2 that is definable by a Σ1-formula with parameter ω2 and has no uncountable homogeneous set. Our results will also allow us to derive tight bounds for the consistency strengths of various partition properties for definable colourings. Finally, we use the developed theory to study the question whether certain homeomorphisms that witness failures of weak compactness at small cardinals can be simply definable.

DERIVED MODELS OF MICE BELOW THE LEAST FIXPOINT OF THE SOLOVAY SEQUENCE

AbstractWe introduce a mouse whose derived model satisfies $AD_ + {\rm{\Theta }} \ge \theta _{\aleph _2 } $. More generally, we will introduce a class of large cardinal properties yielding mice whose derived models can satisfy properties as strong as $AD_ + {\rm{\Theta }} = \theta _{\rm{\Theta }} $.

On the existence of skinny stationary subsets

Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of Pκλ and showed that the existence of skinny stationary subsets of Pκλ is related to large cardinal properties of ideals over Pκλ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

Measurable cardinals and good ‐wellorderings

AbstractWe study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ1‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ? 3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if ? exhibits any of them, with corresponding target ?, then in any forcing extension arising from nontrivial strategically <?-closed forcing $${\mathbb{Q} \in V_\theta}$$Q?V?, the cardinal ? will exhibit none of the large cardinal properties with target ? or larger.

LOCAL CLUB CONDENSATION AND L-LIKENESS

AbstractWe present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.

Large transitive models in local ZFC

This paper is a sequel to Tzouvaras (Arch Math Log 49(5):571–601, 2010), where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and \({\Pi_1^1}\)-indescribable models, were considered. By analogy we refer to such models as “large models”, and the properties in question as “large model properties”. Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, “elementarily embeddable”, “extendible” and “strongly extendible”, “critical” and “strongly critical”, “self-critical” and “strongly self-critical”, the definitions of which involve elementary embeddings. Each large model property ϕ gives rise to a localization axiom Locϕ(ZFC) saying that every set belongs to a transitive model of ZFC satisfying ϕ. The theories LZFCϕ = LZFC + Locϕ(ZFC) are local analogues of the theories ZFC+“there is a proper class of large cardinals ψ”, where ψ is a large cardinal property. If sext(x) is the property of strong extendibility, it is shown that LZFCsext proves Powerset and Σ1-Collection. In order to refute V = L over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom (TMA). V = L can also be refuted by TMA plus the axiom GC saying that “there is a greatest cardinal”, although it is not known if TMA + GC is consistent over LZFC. Finally Vopěnka’s Principle (V P) and its impact on LZFC are examined. It is shown that LZFCsext + V P proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of LZFCsext. Moreover the theories LZFCsext+V P and ZFC+V P are shown to be identical.

Hierarchies of ineffabilities

AbstractWe study combinatorial large cardinal properties on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {P}}_{\kappa } \lambda$\end{document}, such as ineffability, almost ineffability, subtlety, and the Shelah property. We show that, even when λ &gt; κ, the almost ineffability of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {P}}_{\kappa } \lambda$\end{document} does not yield the ineffability of κ. We also show that the Shelah property and the partition property of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {P}}_{\kappa } \lambda$\end{document} do not yield the subtlety of κ.

Guessing models and generalized Laver diamond

We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinal axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinal properties can be defined in terms of suitable elementary embeddings j:Vγ→Vλ. One key observation is that such embeddings are uniquely determined by the image structures j[Vγ]≺Vλ. These structures will be the prototypes guessing models. We shall show, using guessing models M, how to prove for the ordinal κM=jM(crit(jM)) (where πM is the transitive collapse of M and jM is its inverse) many of the combinatorial properties that we can prove for the cardinal j(crit(j)) using the structure j[Vγ]≺Vj(γ). κM will always be a regular cardinal, but consistently can be a successor and guessing models M with κM=ℵ2 exist assuming the proper forcing axiom. By means of these models we shall introduce a new structural property of models of PFA: the existence of a “Laver function” f:ℵ2→Hℵ2 sharing the same features of the usual Laver functions f:κ→Hκ provided by a supercompact cardinal κ. Further applications of our analysis will be proofs of the singular cardinal hypothesis and of the failure of the square principle assuming the existence of guessing models. In particular the failure of square shows that the existence of guessing models is a very strong assumption in terms of large cardinal strength.

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω 2 . Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.

Large cardinals and definable well-orders on the universe

AbstractWe use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

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.

Double helix in large large cardinals and iteration of elementary embeddings

We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals. Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters. As results of our investigation of this helical structure, we have (i) new characterizations of extendible cardinals and of Vopěnkan cardinals in terms of elementary embeddings from the universe V , (ii) a new large large cardinal property which looks like Shelahness properly between Vopěnkaness and almost hugeness, and (iii) a more direct relation between Vopěnkaness and Woodinness than already known. We also consider limits of the helices and relations with the axioms I1–I3.

Orders of Indescribable Sets

We extract some properties of Mahlo’s operation and show that some other very natural operations share these properties. The weakly compact sets form a similar hierarchy as the stationary sets. The height of this hierarchy is a large cardinal property connected to saturation properties of the weakly compact ideal.