We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

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

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary (...) we also see that assuming the existence of a Woodin cardinal δ so that SBHδ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ+ supercompact for all n < ω. We state some results at this level, and indicate how they are proved. (shrink)

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.

We give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further (...) research. (shrink)

We study which [Formula: see text]-distributive forcing notions of size [Formula: see text] can be embedded into tree Prikry forcing notions with [Formula: see text]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [Formula: see text]-distributive forcing notion of size [Formula: see text] be extended to a [Formula: see text]-complete ultrafilter.

If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at N₂ with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's ℙ max axiom (*) holds, then BPFA implies that V is closed under the "Woodin-in-the-next-ZFC-model" operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus "NS ω1 is precipitous" and strengthenings (...) thereof. Along the way, we answer a question of Baumgartner and Taylor. [2, Question 6.11]. (shrink)

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216 89–117] and Steel [PFA implies AD L, J. Symbolic Logic70 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong (...) limit cardinal or PFA. (shrink)

We prove that if image is a Jensen extender model, then image satisfies the Gap-1 morass principle. As a corollary to this and a theorem of Jensen, the model image satisfies the Gap-2 Cardinal Transfer Property → for all infinite cardinals κ and λ.

We prove covering theorems for K, where K is the core model below the sharp for a strong cardinal, and give an application to stationary set reflection.

We present a construction of a global square sequence in extender models with λ-indexing.

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a \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} -sequence if and only if there is a pair of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \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} which, in turn, is equivalent to the existence of a (...) family of size \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} of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \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}. By a result of Burke/jensen, \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} fails whenever \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} is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical \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} -sequence where \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} is the first subcompact. (shrink)

We extend the construction of a global square sequence in extender models from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal k ≥ N₃ such that □k and □(k) fail. 2) There is a cardinal k such that k is weakly compact in the generic extension by Col(k, k⁺). Of special interest is 1) with k = N₃ since it follows from PFA by theorems of (...) Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over $K^c ||k.$. (shrink)

Assume V = L ⊨ ZF + DC + Θ > ω2 + μ is a normal fine measure on.

We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definablecore modelthat is close toVin various ways.

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)

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 $\Sigma _{1}^{2}$ truth for λ is a pair 〈Q, ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ Hλ+ such that (Hλ+; ε, B) $(H_{\lambda +};\in ,B)\vDash \psi [Q]$ . A cardinal λ is $\Sigma _{1}^{2}$ indescribable just in case that for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $\overline{\lambda}<\lambda $ so that $\overline{\lambda}$ is a cardinal and $\langle Q\cap H_{\overline{\lambda}},\psi \rangle $ is a (...) $\Sigma _{1}^{2}$ truth for $\overline{\lambda}$ . More generally, an interval of cardinals [κ, λ] with κ ≤ λ is $\Sigma _{1}^{2}$ indescribable if for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $??\leq \overline{\lambda}<\kappa,??\subseteq H_{\overline{\lambda}}$ , and π: $H_{\overline{\lambda}}\rightarrow H_{\lambda}$ so that $??$ is a cardinal, $\langle ??,\psi \rangle $ is a $\Sigma _{1}^{2}$ truth for $??$ , and π is elementary from $(H_{\overline{\lambda}};\in,??,??)$ with $\pi \,|\,??={\rm id}$ . We prove that the restriction of the proper forcing axiom to c-linked posets requires a $\Sigma _{1}^{2}$ indescribable cardinal in L, and that the restriction of the proper forcing axiom to c⁺-linked posets, in a proper forcing extension of a fine structural model, requires a $\Sigma _{1}^{2}$ indescribable 1-gap [κ, κ⁺]. These results show that the respective forward directions obtained in " Hierarchies of Forcing Axioms I" by Neeman and Schimmerling are optimal. (shrink)