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  1. More on regular and decomposable ultrafilters in ZFC.Paolo Lipparini - 2010 - Mathematical Logic Quarterly 56 (4):340-374.
    We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: If m ≥ 1 and the ultrafilter D is , equation imagem)-regular, then D is κ -decomposable for some κ with λ ≤ κ ≤ 2λ ). If λ is a strong limit cardinal and D is , equation imagem)-regular, then either D is -regular or there are arbitrarily large κ < λ for which D is κ -decomposable ). Suppose that λ is singular, (...)
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  • Decomposable Ultrafilters and Possible Cofinalities.Paolo Lipparini - 2008 - Notre Dame Journal of Formal Logic 49 (3):307-312.
    We use Shelah's theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem: Suppose that $\lambda$ is a singular cardinal, $\lambda ' \lessthan \lambda$, and the ultrafilter $D$ is $\kappa$ -decomposable for all regular cardinals $\kappa$ with $\lambda '\lessthan \kappa \lessthan \lambda$. Then $D$ is either $\lambda$-decomposable or $\lambda ^+$-decomposable. Corollary: If $\lambda$ is a singular cardinal, then an ultrafilter is ($\lambda$,$\lambda$)-regular if and only if it is either $\operator{cf} \lambda$-decomposable or $\lambda^+$-decomposable. We also give applications to (...)
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  • The canonical function game.Paul B. Larson - 2005 - Archive for Mathematical Logic 44 (7):817-827.
    The canonical function game is a game of length ω1 introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ22 absoluteness, cardinality spectra and Π2 maximality for H(ω2) relative to the Continuum Hypothesis.
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  • Bounding by canonical functions, with ch.Paul Larson & Saharon Shelah - 2003 - Journal of Mathematical Logic 3 (02):193-215.
    We show that the members of a certain class of semi-proper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical function for an ordinal less than ω2.
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  • Regular Ultrapowers at Regular Cardinals.Juliette Kennedy, Saharon Shelah & Jouko Väänänen - 2015 - Notre Dame Journal of Formal Logic 56 (3):417-428.
    In earlier work by the first and second authors, the equivalence of a finite square principle $\square^{\mathrm{fin}}_{\lambda,D}$ with various model-theoretic properties of structures of size $\lambda $ and regular ultrafilters was established. In this paper we investigate the principle $\square^{\mathrm{fin}}_{\lambda,D}$—and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, $\square^{\mathrm{fin}}_{\lambda,D}$ holds at regular cardinals for all regular filters $D$ if we assume the generalized continuum hypothesis. In this paper we prove in ZFC that, for certain regular filters (...)
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  • Weak saturation properties and side conditions.Monroe Eskew - 2024 - Annals of Pure and Applied Logic 175 (1):103356.
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  • Covering theorems for the core model, and an application to stationary set reflection.Sean Cox - 2010 - Annals of Pure and Applied Logic 161 (1):66-93.
    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.
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  • On a convenient property about $${[\gamma]^{\aleph_0}}$$.David Asperó - 2009 - Archive for Mathematical Logic 48 (7):653-677.
    Several situations are presented in which there is an ordinal γ such that ${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}$ is a stationary subset of ${[\gamma]^{\aleph_0}}$ for all stationary ${S, T\subseteq \omega_1}$ . A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of ${H_{\omega_2}}$ and the existence of sharps for all reals. Also, an (...)
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  • Incompatible bounded category forcing axioms.David Asperó & Matteo Viale - 2022 - Journal of Mathematical Logic 22 (2).
    Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing (...)
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