We show that o = k++ is necessary for ¬SCH. Together with previous results it provides the exact strenght of ¬SCH.

We characterize the (κ, Λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games $\scr{G}_{<\mu}^{\kappa}(\lambda)$ , when μ ≤ κ ≤ Λ and κ<κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, Λ, μ with μ ≤ Λ, under GCH. In the case when μ ≤ κ ≤ Λ and κ<κ = κ, we show that it is necessary to consider whether the κ-stationarity (...) of Pκ+Λ in the ground model is preserved by B. In this vein, we develop the theory of κ-club and κ-stationary subsets of Pκ+Λ. We also construct Boolean algebras in which Player I wins $\scr{G}_{\kappa}^{\kappa}(\kappa ^{+})$ but the (κ, ∞, κ)-d.1, holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined. (shrink)

Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2 <κ ≤ M(κ, λ), then ◇ κ,λ holds, where M(κ, λ) equals $\lambda ^{\aleph }0$ if cf(λ) ≥ κ, and $(\lambda ^{+})^{\aleph _{0}}$ otherwise.

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: 1. If it is consistent that there is a 1-inaccessible cardinal then it is (...) consistent that ${\sf BC}_{\aleph_1}$.2. If it is consistent that ${\sf BC}_{\aleph_1}$, then it is consistent that there is an inaccessible cardinal.3. If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} + (\forall n < \omega){\sf BC}_{\aleph_n}$ is consistent.4. If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$.5. If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ for a proper class of cardinals $\kappa$ of countable cofinality. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Let$\kappa $be a regular uncountable cardinal, anda cardinal greater than or equal to$\kappa $. Revisiting a celebrated result of Shelah, we show that ifis close to$\kappa $and(= the least size of a cofinal subset of) is greater than, thencan be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that ifand, then no$\kappa $-complete ideal onis weakly-saturated.

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.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Using previous work of Baumgartner, Shelah and others, we describe, for each infinite cardinal θκ, the smallest κ-normal ideal J on Pκ such that.

We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no ideal on Pκ is weakly λ+-saturated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We study normal ideals on Pκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P_{\kappa} }$$\end{document} that are defined in terms of games of uncountable length.

We characterize the -distributive law in Boolean algebras in terms of cut and choose games.

An ideal J on [λ]<κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\lambda ]^{<\kappa }$$\end{document} is said to be [δ]<θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\delta ]^{<\theta }$$\end{document}-normal, where δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} is an ordinal less than or equal to λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, and θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} a cardinal less than or equal (...) to 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}, if given Be∈J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_e \in J$$\end{document} for e∈[δ]<θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e \in [\delta ]^{<\theta }$$\end{document}, the set of all a∈[λ]<κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in [\lambda ]^{<\kappa }$$\end{document} such that a∈Be\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in B_e$$\end{document} for some e∈[a∩δ]<|a∩θ|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e \in [a \cap \delta ]^{< \vert a \cap \theta \vert }$$\end{document} lies in J. We give necessary and sufficient conditions for the existence of such ideals and describe the smallest one, denoted by NSκ,λ[δ]<θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NS_{\kappa,\lambda }^{[\delta ]^{<\theta }}$$\end{document}. We compute the cofinality of NSκ,λ[δ]<θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NS_{\kappa,\lambda }^{[\delta ]^{<\theta }}$$\end{document}. (shrink)