18 found
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  1.  63
    Some results about (+) proved by iterated forcing.Tetsuya Ishiu & Paul B. Larson - 2012 - Journal of Symbolic Logic 77 (2):515-531.
    We shall show the consistency of CH+ᄀ(+) and CH+(+)+ there are no club guessing sequences on ω₁. We shall also prove that ◊⁺ does not imply the existence of a strong club guessing sequence ω₁.
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  2.  29
    Iterated elementary embeddings and the model theory of infinitary logic.John T. Baldwin & Paul B. Larson - 2016 - Annals of Pure and Applied Logic 167 (3):309-334.
  3.  53
    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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  4.  22
    Almost galois ω-stable classes.John T. Baldwin, Paul B. Larson & Saharon Shelah - 2015 - Journal of Symbolic Logic 80 (3):763-784.
  5.  43
    Martin’s Maximum and definability in H.Paul B. Larson - 2008 - Annals of Pure and Applied Logic 156 (1):110-122.
    In [P. Larson, Martin’s Maximum and the axiom , Ann. Pure App. Logic 106 135–149], we modified a coding device from [W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co, Berlin, 1999] and the consistency proof of Martin’s Maximum from [M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annal. Math. 127 1–47] to show that from a supercompact limit of supercompact cardinals one could force Martin’s (...)
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  6.  24
    Maximal Tukey types, P-ideals and the weak Rudin–Keisler order.Konstantinos A. Beros & Paul B. Larson - 2023 - Archive for Mathematical Logic 63 (3):325-352.
    In this paper, we study some new examples of ideals on $$\omega $$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic (...)
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  7.  36
    The stationary set splitting game.Paul B. Larson & Saharon Shelah - 2008 - Mathematical Logic Quarterly 54 (2):187-193.
    The stationary set splitting game is a game of perfect information of length ω1 between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ22 maximality with a predicate for the nonstationary ideal on ω1, and an example of a consistently (...)
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  8.  21
    Foundations of Mathematics.Andrés Eduardo Caicedo, James Cummings, Peter Koellner & Paul B. Larson (eds.) - 2016 - American Mathematical Society.
    This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...)
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  9.  68
    ℙmax variations for separating club guessing principles.Tetsuya Ishiu & Paul B. Larson - 2012 - Journal of Symbolic Logic 77 (2):532-544.
    In his book on P max [7], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω | . In this paper we employ one of the techniques from this book to produce P max variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [4] while studying games of length ω | . It was shown in [1] that (...)
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  10.  42
    Regular embeddings of the stationary tower and Woodin's Σ 2 2 maximality theorem.Richard Ketchersid, Paul B. Larson & Jindřich Zapletal - 2010 - Journal of Symbolic Logic 75 (2):711-727.
    We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V δ . We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding (...)
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  11.  21
    Coding with canonical functions.Paul B. Larson & Saharon Shelah - 2017 - Mathematical Logic Quarterly 63 (5):334-341.
    A function f from ω1 to the ordinals is called a canonical function for an ordinal α if f represents α in any generic ultrapower induced by forcing with math formula. We introduce here a method for coding sets of ordinals using canonical functions from ω1 to ω1. Combining this approach with arguments from, we show, assuming the Continuum Hypothesis, that for each cardinal κ there is a forcing construction preserving cardinalities and cofinalities forcing that every subset of κ is (...)
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  12.  29
    Proper Forcings and Absoluteness in LProper Forcing and L.Paul B. Larson, Itay Neeman & Jindrich Zapletal - 2002 - Bulletin of Symbolic Logic 8 (4):548.
  13.  38
    Three Days of Ω-logic( Mathematical Logic and Its Applications).Paul B. Larson - 2011 - Annals of the Japan Association for Philosophy of Science 19:57-86.
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  14.  38
    The Filter dichotomy and medial limits.Paul B. Larson - 2009 - Journal of Mathematical Logic 9 (2):159-165.
    The Filter Dichotomy says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A medial limit is a universally measurable function from [Formula: see text] to the unit interval [0, 1] which is finitely additive for disjoint sets, and maps singletons to 0 and ω to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. (...)
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  15.  68
    The Nonstationary Ideal in the Pmax Extension.Paul B. Larson - 2007 - Journal of Symbolic Logic 72 (1):138 - 158.
    The forcing construction Pmax, invented by W. Hugh Woodin, produces a model whose collection of subsets of ω₁ is in some sense maximal. In this paper we study the Boolean algebra induced by the nonstationary ideal on ω₁ in this model. Among other things we show that the induced quotient does not have a simply definable form. We also prove several results about saturation properties of the ideal in this extension.
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  16.  27
    (2 other versions)Games of countable length. Sets and Proofs. [REVIEW]Paul B. Larson - 2005 - Bulletin of Symbolic Logic 11 (4):542-544.
  17.  34
    Itay Neeman and Jindřich Zapletal. Proper forcings and absoluteness in L. Commentationes mathematicae Universitatis Carolinae, vol. 39 , pp. 281–301. - Itay Neeman and Jindřich Zapletal. Proper forcing and L. The journal of symbolic logic, vol. 66 , pp. 801–810. [REVIEW]Paul B. Larson - 2002 - Bulletin of Symbolic Logic 8 (4):548-550.
  18.  36
    (1 other version)W. Hugh Woodin. The axiom of determinacy, forcing axioms, and the nonstationary ideal. De Gruyter series in logic and its applications, no. 1. Walter de Gruyter, Berlin and New York 1999, vi + 934 pp. [REVIEW]Paul B. Larson - 2002 - Bulletin of Symbolic Logic 8 (1):91-93.