30 found
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  1. The Consistency Strength of the Free-Subset Property for Ωω.Peter Koepke - 1984 - Journal of Symbolic Logic 49 (4):1198 - 1204.
  2.  22
    Register Computations on Ordinals.Peter Koepke & Ryan Siders - 2008 - Archive for Mathematical Logic 47 (6):529-548.
    We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register (...)
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  3.  24
    Turing Computations on Ordinals.Peter Koepke - 2005 - Bulletin of Symbolic Logic 11 (3):377-397.
    We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel's constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L.
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  4.  14
    The Basic Theory of Infinite Time Register Machines.Merlin Carl, Tim Fischbach, Peter Koepke, Russell Miller, Miriam Nasfi & Gregor Weckbecker - 2010 - Archive for Mathematical Logic 49 (2):249-273.
    Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set (...)
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  5.  7
    On the Consistency Strength of ‘Accessible’ Jonsson Cardinals and of the Weak Chang Conjecture.Hans-Dieter Donder & Peter Koepke - 1983 - Annals of Pure and Applied Logic 25 (3):233-261.
  6.  3
    Ordinal Machines and Admissible Recursion Theory.Peter Koepke & Benjamin Seyfferth - 2009 - Annals of Pure and Applied Logic 160 (3):310-318.
    We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the (...)
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  7.  15
    Towards a Theory of Infinite Time Blum-Shub-Smale Machines.Peter Koepke & Benjamin Seyfferth - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 405--415.
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  8.  13
    An Elementary Approach to the Fine Structure of L.Sy D. Friedman & Peter Koepke - 1997 - Bulletin of Symbolic Logic 3 (4):453-468.
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  9.  37
    The Category of Inner Models.Peter Koepke - 2002 - Synthese 133 (1-2):275 - 303.
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  10.  21
    Making All Cardinals Almost Ramsey.Arthur W. Apter & Peter Koepke - 2008 - Archive for Mathematical Logic 47 (7-8):769-783.
    We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...)
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  11.  2
    Some Applications of Short Core Models.Peter Koepke - 1988 - Annals of Pure and Applied Logic 37 (2):179-204.
    We survey the definition and fundamental properties of the family of short core models, which extend the core model K of Dodd and Jensen to include α-sequences of measurable cardinals . The theory is applied to various combinatorial principles to get lower bounds for their consistency strengths in terms of the existence of sequences of measurable cardinals. We consider instances of Chang's conjecture, ‘accessible’ Jónsson cardinals, the free subset property for small cardinals, a canonization property of ω ω , and (...)
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  12.  6
    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.
    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 (...)
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  13.  9
    Superatomic Boolean Algebras Constructed From Morasses.Peter Koepke & Juan Carlos Martínez - 1995 - Journal of Symbolic Logic 60 (3):940-951.
    By using the notion of a simplified (κ,1)-morass, we construct κ-thin-tall, κ-thin-thick and, in a forcing extension, κ-very thin-thick superatomic Boolean algebras for every infinite regular cardinal κ.
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  14.  5
    Homogeneously Souslin Sets in Small Inner Models.Peter Koepke & Ralf Schindler - 2006 - Archive for Mathematical Logic 45 (1):53-61.
    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.
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  15.  19
    On the Elimination of Malitz Quantifiers Over Archimedian Real Closed Fields.Peter Koepke - 1989 - Archive for Mathematical Logic 28 (3):167-171.
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  16.  9
    On the Free Subset Property at Singular Cardinals.Peter Koepke - 1989 - Archive for Mathematical Logic 28 (1):43-55.
    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.
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  17.  1
    Regularity of Ultrafilters and the Core Model.Hans-Dieter Donder, Peter Koepke, Jean-Pierre Levinski & D. J. Walker - 1990 - Journal of Symbolic Logic 55 (3):1313-1315.
  18.  13
    Hyperfine Structure Theory and Gap 1 Morasses.Sy-David Friedman, Peter Koepke & Boris Piwinger - 2006 - Journal of Symbolic Logic 71 (2):480 - 490.
    Using the Friedman-Koepke Hyperfine Structure Theory of [2], we provide a short construction of a gap 1 morass in the constructible universe.
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  19.  32
    The Consistency Strength of Choiceless Failures of SCH.Arthur W. Apter & Peter Koepke - 2010 - Journal of Symbolic Logic 75 (3):1066-1080.
    We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...)
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  20.  18
    A Minimal Prikry-Type Forcing for Singularizing a Measurable Cardinal.Peter Koepke, Karen Räsch & Philipp Schlicht - 2013 - Journal of Symbolic Logic 78 (1):85-100.
    Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality (...)
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  21.  7
    Extenders, Embedding Normal Forms, and the Martin-Steel-Theorem.Peter Koepke - 1998 - Journal of Symbolic Logic 63 (3):1137-1176.
    We propose a simple notion of "extender" for coding large elementary embeddings of models of set theory. As an application we present a self-contained proof of the theorem by D. Martin and J. Steel that infinitely many Woodin cardinals imply the determinacy of every projective set.
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  22. X1. Introduction. In 1938, K. Gödel Defined the Model L of Set Theory to Show the Relative Consistency of Cantor's Continuum Hypothesis. L is Defined as a Union L=. [REVIEW]Syd Friedman & Peter Koepke - 1997 - Bulletin of Symbolic Logic 3 (4).
  23.  3
    The First Measurable Cardinal Can Be the First Uncountable Regular Cardinal at Any Successor Height.Arthur W. Apter, Ioanna M. Dimitriou & Peter Koepke - 2014 - Mathematical Logic Quarterly 60 (6):471-486.
  24.  1
    Global Square and Mutual Stationarity at the ℵn.Peter Koepke & Philip D. Welch - 2011 - Annals of Pure and Applied Logic 162 (10):787-806.
    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 (...)
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  25.  1
    The Consistency Strength of the Free-Subset Property for $Omega_omega$.Peter Koepke - 1984 - Journal of Symbolic Logic 49 (4):1198-1204.
  26. The Consistency Strength of InlineEquation ID=" IEq1"> EquationSource Format=" TEX"> ImageObject Color=" BlackWhite" FileRef=" 15320065ArticleIEq1. Gif" Format=" GIF" Rendition=" HTM" Type=" Linedraw"/> And. [REVIEW]Arthur W. Apter & Peter Koepke - 2006 - Archive for Mathematical Logic 45 (6):721-738.
  27. On the Consistency Strength of `Accessible' Jonsson Cardinals and of the Weak Chang Conjecture.Some Applications of Short Core Models.Sy D. Friedman, Hans-Dieter Donder & Peter Koepke - 1989 - Journal of Symbolic Logic 54 (4):1496.
  28. To Show the Relative Consistency of Cantor's Continuum Hypothesis. L is Defined as a Union L=⋃.Syd Friedman & Peter Koepke - 1997 - Bulletin of Symbolic Logic 3 (4).
  29. Extenders, Embedding Normal Forms, and the Martin-Steel-Theorem.Peter Koepke - 1998 - Journal of Symbolic Logic 63 (3):1137-1176.
    We propose a simple notion of "extender" for coding large elementary embeddings of models of set theory. As an application we present a self-contained proof of the theorem by D. Martin and J. Steel that infinitely many Woodin cardinals imply the determinacy of every projective set.
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  30. Covering Properties of Core Models.Ernest Schimmerling, Peter Koepke, William J. Mitchell & John R. Steel - 2004 - Bulletin of Symbolic Logic 10 (4):583-588.
     
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