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Some applications of short core models

Peter Koepke

Annals of Pure and Applied Logic 37 (2):179-204 (1988)

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Measurable cardinals and good ‐wellorderings.Philipp Lücke & Philipp Schlicht - 2018 - Mathematical Logic Quarterly 64 (3):207-217.details We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ1‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows that (...) Download Export citation Bookmark 2 citations
Homogeneously Souslin sets in small inner models.Peter Koepke & Ralf Schindler - 2006 - Archive for Mathematical Logic 45 (1):53-61.details We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0long does not exist, or else (b) V = K, where K is the core model below a μ-measurable cardinal. Download Export citation Bookmark 1 citation
The Consistency Strength of $$\aleph{\omega}$$ and $$\aleph{{\omega}_1}$$ Being Rowbottom Cardinals Without the Axiom of Choice.Arthur W. Apter & Peter Koepke - 2006 - Archive for Mathematical Logic 45 (6):721-737.details We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. We (...) Download Export citation Bookmark 4 citations
Co-stationarity of the Ground Model.Natasha Dobrinen & Sy-David Friedman - 2006 - Journal of Symbolic Logic 71 (3):1029 - 1043.details This paper investigates when it is possible for a partial ordering P to force Pκ(λ) \ V to be stationary in VP. It follows from a result of Gitik that whenever P adds a new real, then Pκ(λ) \ V is stationary in VP for each regular uncountable cardinal κ in VP and all cardinals λ > κ in VP [4]. However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The (...) Download Export citation Bookmark 3 citations
More canonical forms and dense free subsets.Heike Mildenberger - 2004 - Annals of Pure and Applied Logic 125 (1-3):75-99.details Assuming the existence of ω compact cardinals in a model on GCH, we prove the consistency of some new canonization properties on ω. Our aim is to get as dense patterns in the distribution of indiscernibles as possible. We prove Theorem 2.1. thm2.1Suppose the consistency of “ZFC+GCH + there are infinitely many compact cardinals”. Then the following is consistent: ZFC+GCH + and for every family 0 (...) such that for all 0shrink) Download Export citation Bookmark 2 citations
On the free subset property at singular cardinals.Peter Koepke - 1989 - Archive for Mathematical Logic 28 (1):43-55.details We give a proof ofTheorem 1. Let κ be the smallest cardinal such that the free subset property Fr ω (κ,ω 1)holds. Assume κ is singular. Then there is an inner model with ω1 measurable cardinals. Download Export citation Bookmark 3 citations
Order types of free subsets.Heike Mildenberger - 1997 - Annals of Pure and Applied Logic 89 (1):75-83.details We give for ordinals α a lower bound for the least ordinal α such that Frordξ,β) and show that given enough measurable cardinals there are forcing extensions where the given bounds are sharp. Download Export citation Bookmark 1 citation

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