We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that $(I[\mu ^{ + + } ])^V $ generates I[µ⁺⁺] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

We construct a model where every increasing ω-sequence of regular cardinals carries a mutually stationary sequence which is not tightly stationary, and show that this property is preserved under a class of Prikry-type forcings. Along the way, we give examples in the Cohen and Prikry models of ω-sequences of regular cardinals for which there is a non-tightly stationary sequence of stationary subsets consisting of cofinality ω_1 ordinals, and show that such stationary sequences are mutually stationary in the presence of interleaved (...) supercompact cardinals. (shrink)

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

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 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.

We construct a model in which for all 1 ≤ n < ω, there is no stationary subset of ${\aleph_{n+1} \cap {\rm cof}(\aleph_n)}$ which carries a partial square.

From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis fails, there is a bad scale at κ and κ ++ has the tree property. In particular this model has no special κ +-trees.

We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(\mu^+,\mu^+,\mu^+,cf(\mu))$ for singular $\mu$.

We describe a recipe for generating normal ideals on successors of singular cardinals. We show that these ideals are related to many weakenings of □ that have appeared in the literature. Our main purpose, however, is to provide an organized list of open questions related to these ideals.

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)

A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinalκsatisfies thenarrow system propertyif every narrow system of heightκhas a cofinal branch. In this paper, we study connections between the narrow system property, square principles, and forcing axioms. We prove, assuming large cardinals, both that it is consistent that ℵω+1satisfies the narrow system property and$\square _{\aleph _\omega, < \aleph (...) _\omega } $holds and that it is consistent that every regular cardinal satisfies the narrow system property. We introduce natural strengthenings of classical square principles and show how they can be used to produce narrow systems with no cofinal branch. Finally, we show that the Proper Forcing Axiom implies that every narrow system of countable width has a cofinal branch but is consistent with the existence of a narrow system of width ω1with no cofinal branch. (shrink)

We prove that the Weak Reflection Principle does not imply that every stationary set reflects to an internally approachable set. We show that several variants of internal approachability, namely, internally unbounded, internally stationary, and internally club, are not provably equivalent.

We 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. (shrink)

We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal $\mu$, we have $\mu^+\nrightarrow[\mu^+]^2_{\mu^+}$ if and only if $\mu^+\nrightarrow[\mu^+]^2_{\theta}$ for arbitrarily large $\theta < \mu$.

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.