Regular Cardinals Research Articles

Unavoidable structures in infinite tournaments

We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs G G , either (i) there is a countably-infinite tournament K K such that G ⊈ K G\not \subseteq K , or (ii) every countably-infinite tournament contains a spanning copy of G G . Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals κ \kappa ; however, we are only able to provide a concise characterization in the case when κ = ℵ 1 \kappa =\aleph _1 .

Narrow systems revisited

AbstractWe investigate connections between set‐theoretic compactness principles and cardinal arithmetic, introducing and studying generalized narrow system properties as a way to approach two open questions about two‐cardinal tree properties. The first of these questions asks whether the strong tree property at a regular cardinal implies the singular cardinals hypothesis () above . We show here that a certain narrow system property at that is closely related to the strong tree property, and holds in all known models thereof, suffices to imply above . The second of these questions asks whether the strong tree property can consistently hold simultaneously at all regular cardinals . We show here that the analogous question about the generalized narrow system property has a positive answer. We also highlight some connections between generalized narrow system properties and the existence of certain strongly unbounded subadditive colorings.

Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products

We investigate two different ways of recovering a Grothendieck category as a filtered bicolimit of small categories and the compatibility of both with the tensor product of Grothendieck categories. Firstly, we show that any locally presentable linear category (and in particular any Grothendieck category) can be recovered as the filtered bicolimit of its subcategories of α-presentable objects, with α varying in the family of small regular cardinals. We then prove that the tensor product of locally presentable linear categories (and in particular the tensor product of Grothendieck categories) can be recovered as a fltered bicolimit of the Kelly tensor product of α-cocomplete linear categories of the corresponding subcategories of α-presentable objects. Secondly, we show that one can recover any Grothendieck category as a filtered bicolimit of its linear site presentations. We then prove that the tensor product of Grothendieck categories, in contrast with the first case, cannot be recovered in general as a filtered bicolimit of the tensor product of the corresponding linear sites. Finally, we show how the first presentation can be helpful when computing tensor products of cocontinuous linear functors between Grothendieck categories.

Approachable free subsets and fine structure derived scales

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders {o(μ)|μ<λ} is unbounded in λ. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

LOGICS FROM ULTRAFILTERS

Abstract Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $ -closed logic ${\mathcal {L}}_\Omega $ . ${\mathcal {L}}_\Omega $ is $\omega $ -relatively compact iff some $D\in \Omega $ fails to be $\omega _1$ -complete iff ${\mathcal {L}}_\Omega $ does not contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$ , if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $ , then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers

We prove that the theory of the models constructible using finitely many cofinality quantifiers – Cλ1,…,λn⁎ and C<λ1,…,<λn⁎ for λ1,…,λn regular cardinals – is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to <μ-closed sets.

STRUCTURAL PROPERTIES OF THE STABLE CORE

AbstractThe stable core, an inner model of the form $\langle L[S],\in , S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in V, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots , C_n]$ for specially nested class clubs $C_1, \dots , C_n$ , like those arising in the stable core, generalizing recent results of Welch [29].

On notions of compactness, object classifiers, and weak Tarski universes

AbstractWe prove a correspondence between $\kappa$ -small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$ -compact maps in their underlying quasi-categories for suitably large regular cardinals $\kappa$ . We thus obtain a transition result between weakly universal small fibrations in the (type-theoretic) injective Dugger–Rezk-style standard presentations of model toposes and object classifiers in Grothendieck $\infty$ -toposes in the sense of Lurie.

The double density spectrum of a topological space

It is an interesting, maybe surprising, fact that different dense subspaces of even “nice” topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space X that we call the double density spectrum of X and denote by dd(X).We improve a result from [1] by showing that dd(X) is always ω-closed (i.e., countably closed) if X is Hausdorff.We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows.Let S be a non-empty set of infinite cardinals. Then (1) S = dd(X) holds for a Hausdorff space X iff S is ω-closed and \(\sup \,S \le {2^{{2^{\min S}}}}\) (2) S = dd(X) holds for a regular space X iff S is ω-closed and sup S ≤ 2min S. We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance: (i) If κ = cf(κ)embeds in \({\cal P}\left( \omega \right)/{\rm{fin}}\) and S is any set of uncountable regular cardinals < κ with ∣S∣ < min S, then there is a compactum C such that {ω, κ}∪ S ⊂ dd(C), moreover λ ∉ dd(C) whenever ∣S∣ + ω < cf(λ) < κ and cf(λ) ∉ S. (ii) It is consistent to have a separable compactum C such that dd(C) is not ω1-closed.

Controlling the number of normal measures at successor cardinals

AbstractWe examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.

TREES AND STATIONARY REFLECTION AT DOUBLE SUCCESSORS OF REGULAR CARDINALS

AbstractWe obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa $ , updating some classical constructions in the process. This includes models of $\mathsf {CSR}(\kappa ^{++})\wedge {\sf TP}(\kappa ^{++})$ (both with and without ${\sf AP}(\kappa ^{++})$ ) and models of the conjunctions ${\sf SR}(\kappa ^{++}) \wedge \mathsf {wTP}(\kappa ^{++}) \wedge {\sf AP}(\kappa ^{++})$ and $\neg {\sf AP}(\kappa ^{++}) \wedge {\sf SR}(\kappa ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author [9], and is here given using different methods). Analogs of these results with the failure of $\sf {SH}(\kappa ^{++})$ are given as well. Finally, we obtain all of our results with an arbitrarily large $2^\kappa $ , applying recent joint work by Honzik and the third author.

A class of higher inductive types in Zermelo‐Fraenkel set theory

AbstractWe define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo‐Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.

Towers and gaps at uncountable cardinals

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(\kap

Prikry-type forcing and the set of possible cofinalities

It is known that the set of possible cofinalities pcf($A$) has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that pcf($A$) has no good properties in the presence of a measurable cardinal, or in generic extensions by Prikry-type forcing.

MORE ON THE PRESERVATION OF LARGE CARDINALS UNDER CLASS FORCING

AbstractWe prove two general results about the preservation of extendible and $C^{(n)}$ -extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{(n)}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{(n)}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$ -definable behaviour of the power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving $C^{(n)}$ -extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings–Foreman–Magidor for forcing $\diamondsuit _{\kappa ^+}^+$ at every $\kappa $ [10] preserves $C^{(n)}$ -extendible cardinals. We give an optimal result on the consistency of weak square principles and $C^{(n)}$ -extendible cardinals. In the last section prove another preservation result for $C^{(n)}$ -extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations. As applications we prove the consistency of $C^{(n)}$ -extendible cardinals with $\mathrm {{V}}=\mathrm {{HOD}}$ , and also with $\mathrm {GA}$ (the Ground Axiom) plus $\mathrm {V}\neq \mathrm {HOD}$ , the latter being a strengthening of a result from [14].

Colouring of successor of regular, again

We get a version of the colouring property Pr1 proving \(\Pr_1(\lambda,\lambda,\lambda,\partial)\) always when \(\lambda= \partial^+,\partial \) are regular cardinals and some stationary subset of \(\lambda\) consisting of ordinals of cofinality \(< \partial\) do not reflect in any ordinal \(< \lambda\).

MORE ZFC INEQUALITIES BETWEEN CARDINAL INVARIANTS

AbstractMotivated by recent results and questions of Raghavan and Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $\kappa =\lambda ^+$ for some $\lambda \geq \omega $ and $\mathfrak {b}(\kappa )=\kappa ^+$ then $\mathfrak {a}_e(\kappa )=\mathfrak {a}_p(\kappa )=\kappa ^+$ . If, additionally, $2^{&lt;\lambda }=\lambda $ then $\mathfrak {a}_g(\kappa )=\kappa ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak {d}(\kappa )$ in terms of $\mathfrak {r}(\kappa )$ , including $\mathfrak {d}(\kappa )\leq \mathfrak {r}_\sigma (\kappa )\leq \operatorname {\mathrm {cf}}([\mathfrak {r}(\kappa )]^\omega )$ , and $\mathfrak {d}(\kappa )\leq \mathfrak {r}(\kappa )$ whenever $\mathfrak {r}(\kappa )&lt;\mathfrak {b}(\kappa )^{+\kappa }$ or $\operatorname {\mathrm {cf}}(\mathfrak {r}(\kappa ))\leq \kappa $ holds.

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

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 “ $(\lambda ^{+\omega })^{\mathrm {HOD}}&lt;\lambda ^+$ , 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.

Representability and compactness for pseudopowers

We prove a compactness theorem for pseudopower operations of the form $${{\,\mathrm{pp}\,}}_{\Gamma (\mu ,\sigma )}(\mu )$$ where $$\aleph _0<\sigma ={{\,\mathrm{cf}\,}}(\sigma )\le {{\,\mathrm{cf}\,}}(\mu )$$ . Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which $${{\,\mathrm{pcf}\,}}(A)$$ has an inaccessible accumulation point.

Successive failures of approachability

Motivated by showing that in ZFC we cannot construct a special Aronszajn tree on some cardinal greater than \({\aleph _1}\), we produce a model in which the approachability property fails (hence there are no special Aronszajn trees) at all regular cardinals in the interval \(\left[ {{\aleph _2},{\aleph _{{\omega ^2} + 3}}} \right]\) and \({{\aleph _{{\omega ^2}}}}\) is strong limit.