We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at $\aleph_{\omega + 1}$ . This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor—Shelah [7].

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove (...) from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ. (shrink)

We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor—Shelah [7].

We show that given ù many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω²+1 and the SCH fails at ℵ ω².

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection (...) of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems. Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras. The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004. (shrink)

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

We address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik–Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik–Sharon model and other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales.

We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ) equal to κ++, and use it to prove the following theorem on mutual stationarity at n.Let ω1 denote the first uncountable cardinal of V and set to be the class of ordinals of cofinality ω1.TheoremIf every sequence n m. In particular, there is such a model in which for (...) all sufficiently large m<ω, the class of measurables λ with oM≥ωm is, in V, stationary below m+2. (shrink)

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)

We improve on results and constructions by Apter, Dimitriou, Gitik, Hayut, Karagila, and Koepke concerning large cardinals, ultrafilters, and cofinalities without the axiom of choice. In particular, we show the consistency of the following statements from certain assumptions: the first supercompact cardinal can be the first uncountable regular cardinal, all successors of regular cardinals are Ramsey, every sequence of stationary sets in is mutually stationary, an infinitary Chang conjecture holds for the cardinals, and all are singular. In each of the (...) cases, our results either weaken the hypotheses or strengthen the conclusions of known proofs. (shrink)

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of $$\textsf {ZFC}$$ ZFC, then $$\textsf {DC}_{<\kappa }$$ DC < κ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is $$\kappa $$ (...) κ -distributive and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. Further we observe that if $$\delta <\kappa $$ δ < κ and V is a model of $$\textsf {ZF}+\textsf {DC}_{\delta }$$ ZF + DC δ, then $$\textsf {DC}_{\delta }$$ DC δ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is ($$\delta +1$$ δ + 1 )-strategically closed and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. (shrink)