We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the (...) existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. (shrink)

We give a construction of the square principle by means of forcing with finite conditions.

If are such that δ is indestructibly supercompact and γ is measurable, then it must be the case that level by level inequivalence between strong compactness and supercompactness fails. We prove a theorem which points to this result being best possible. Specifically, we show that relative to the existence of cardinals such that κ1 is λ‐supercompact and λ is inaccessible, there is a model for level by level inequivalence between strong compactness and supercompactness containing a supercompact cardinal in which κ’s (...) strong compactness, but not supercompactness, is indestructible under κ‐directed closed forcing. In this model, κ is the least strongly compact cardinal, and no cardinal is supercompact up to an inaccessible cardinal. (shrink)

For sentences $\phi$ of $L_{\omega_{1},\omega}$, we investigate the question of absoluteness of $\phi$ having models in uncountable cardinalities. We first observe that having a model in $\aleph_{1}$ is an absolute property, but having a model in $\aleph_{2}$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis context and provide sentences for any $\alpha\in\omega_{1}\setminus\{0,1,\omega\}$ for which the existence of a model in $\aleph_{\alpha}$ is nonabsolute . Finally, we present a complete (...) sentence for which model existence in $\aleph_{3}$ is nonabsolute. (shrink)

We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.

We 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]. (shrink)

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if, then (a) is not (λ, 2)‐distributive, and (b) does not hold.

We show the consistency, relative to the appropriate supercompactness or strong compactness assumptions, of the existence of a non-supercompact strongly compact cardinal $\kappa _0$ (the least measurable cardinal) exhibiting properties which are impossible when $\kappa _0$ is supercompact. In particular, we construct models in which $\square _{\kappa ^+}$ holds for every inaccessible cardinal $\kappa $ except $\kappa _0$, GCH fails at every inaccessible cardinal except $\kappa _0$, and $\kappa _0$ is less than the least Woodin cardinal.

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on [Formula: see text], if [Formula: see text] is weakly compact, then [Formula: see text] holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails (...) at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [O. B. -Neria, Diamonds, compactness, and measure sequences, J. Math. Log. 19(1) (2019) 1950002]. (shrink)

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of freshness and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the (...) other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties. (shrink)

We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.

We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a ideal J extending the nonstationary ideal on a regular uncountable cardinal \, our goal being to witness the nonsaturation of J by the existence of towers ).

In this paper we study the notion of $C^{}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{}$ -supercompact but also that the least supercompact is $C^{}$ -supercompact }$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong (...) coloring principle U(. . .), introduced in Part I of this series. (shrink)

We show that the consistency of the theories “ holds below ” + “there is an injection ” + “both and fail” and + “ holds below ” + “there is an injection ” + “ satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long‐standing open questions concerning, weak square, approachability, and the tree property. There is nothing special about, and the injection into can be from any ordinal λ (...) (which means that pcf theory can fail badly in the models constructed if ). (shrink)

We show that it is consistent relative to a huge cardinal that for all infinite cardinals [Formula: see text], [Formula: see text] holds and there is a stationary [Formula: see text] such that [Formula: see text] is [Formula: see text]-saturated.

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height ω1 has a nonspecial subtree of size ℵ1. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of PFA, which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible with the Baire (...) Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations. (shrink)

An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree. In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

We construct a model [Formula: see text] of [Formula: see text] which lies between [Formula: see text] and [Formula: see text] for a Cohen real [Formula: see text] and does not have the form [Formula: see text] for any set [Formula: see text]. This is loosely based on the unwritten work done in a Bristol workshop about Woodin’s HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking (...) us one step closer to understanding models of [Formula: see text], and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of [Formula: see text], and we use it to show the failure of Kinna–Wagner Principles in [Formula: see text]. (shrink)

The effects of the forcing axioms \, \ and \ on the failure of weak threaded square principles of the form \\) are analyzed. To this end, a diagonal reflection principle, \, and it implies the failure of \\) if \. It is also shown that this result is sharp. It is noted that \/\ imply the failure of \\), for every regular \, and that this result is sharp as well.

Shelah and Stanley :887–897, 1988) constructed a κ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa ^+$$\end{document}-Aronszjan tree with an ascent path using □κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\kappa }$$\end{document}. We show that □κ,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\kappa,2}$$\end{document} does not imply the existence of Aronszajn trees with ascent paths. The proof goes through an intermediate combinatorial principle, which we investigate further.

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of introduced by Brodsky and Rinot for the purpose of constructing κ‐Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at κ but the stronger is not. We then prove that, if μ is a singular cardinal, implies the existence of a special ‐tree with a cf(μ)‐ascent path, thus answering a question of Lücke.

We investigate the compatibility of $$I_0$$ with various combinatorial principles at $$\lambda ^+$$, which include the existence of $$\lambda ^+$$ -Aronszajn trees, square principles at $$\lambda $$, the existence of good scales at $$\lambda $$, stationary reflections for subsets of $$\lambda ^{+}$$, diamond principles at $$\lambda $$ and the singular cardinal hypothesis at $$\lambda $$. We also discuss whether these principles can hold in $$L(V_{\lambda +1})$$.

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...) show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ℵ2 and arrange in the generic extension that simultaneous reflection holds at ℵ2, and at the same time, every stationary subset of ℵ3 concentrating on points of cofinality ω has a reflection point of cofinality ω1. (shrink)

Our main results are: (A) It is consistent relative to a large cardinal that holds but fails. (B) If holds and are two infinite cardinals such that and λ carries a good scale, then holds. (C) If are two cardinals such that κ is λ‐Shelah and, then there is no good scale for λ.

We present several forcing posets for adding a non-reflecting stationary subset of Pω1, where λ≥ω2. We prove that PFA is consistent with dense non-reflection in Pω1, which means that every stationary subset of Pω1 contains a stationary subset which does not reflect to any set of size 1. If λ is singular with countable cofinality, then dense non-reflection in Pω1 follows from the existence of squares.

We construct a model in which the singular cardinal hypothesis fails at ℵωℵω. We use characterizations of genericity to show the existence of a projection between different Prikry type forcings.

We introduce the weak partial square principles View the MathML source and View the MathML source, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that □λ,κ does not imply View the MathML source.

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof and subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between □κ and □ for a regular cardinal κ. We provide a detailed (...) picture of the implications between these square principles. (shrink)

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

We construct two models in which the least strongly compact cardinal κ is also the least strong cardinal. In each of these models, κ satisfies indestructibility properties for both its strong compactness and strongness.

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at κ then every stationary subset of reflects. It will also be demonstrated (...) that MRP always fails in a generic extension by Prikry forcing. (shrink)

We study a normal ideal on Pκ that is defined in terms of games.

We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. In particular, the simplest of our results establishes that if μ is singular and , then there is a regular cardinal θ<μ such that any fewer than cf stationary subsets of must reflect simultaneously.