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Paul Larson [21]Paul B. Larson [20]
  1.  37
    An Variation for One Souslin Tree.Paul Larson - 1999 - Journal of Symbolic Logic 64 (1):81-98.
    We present a variation of the forcing S max as presented in Woodin [4]. Our forcing is a P max -style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree T G which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with T G being this minimal tree. In (...)
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  2.  33
    Separating Stationary Reflection Principles.Paul Larson - 2000 - Journal of Symbolic Logic 65 (1):247-258.
    We present a variety of (ω 1 ,∞)-distributive forcings which when applied to models of Martin's Maximum separate certain well known reflection principles. In particular, we do this for the reflection principles SR, SR α (α ≤ ω 1 ), and SRP.
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  3.  10
    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.
  4.  26
    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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  5.  31
    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 relative to the Continuum Hypothesis.
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  6.  26
    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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  7.  3
    Almost Galois Ω-Stable Classes.John T. Baldwin, Paul B. Larson & Saharon Shelah - 2015 - Journal of Symbolic Logic 80 (3):763-784.
  8.  29
    The Size of $\tilde{T}$.Paul Larson - 2000 - Archive for Mathematical Logic 39 (7):541-568.
    Given a stationary subset T of $\omega_{1}$ , let $\tilde{T}$ be the set of ordinals in the interval $(\omega_{1}, \omega_{2})$ which are necessarily in the image of T by any embedding derived from the nonstationary ideal. We consider the question of the size of $\tilde{T}$ , givenT, and use Martin's Maximum and $\mathbb{P}_{max}$ to give some answers.
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  9.  6
    Canonical Models for Fragments of the Axiom of Choice.Paul Larson & Jindřich Zapletal - 2017 - Journal of Symbolic Logic 82 (2):489-509.
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  10.  23
    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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  11.  15
    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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  12.  14
    Increasing Δ 1 2 and Namba-Style Forcing.Richard Ketchersid, Paul Larson & Jindřich Zapletal - 2007 - Journal of Symbolic Logic 72 (4):1372-1378.
    We isolate a forcing which increases the value of δ12 while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.
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  13.  18
    Saturation, Suslin Trees and Meager Sets.Paul Larson - 2005 - Archive for Mathematical Logic 44 (5):581-595.
    We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω 1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary (...)
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  14.  23
    Three Days of Ω-Logic.Paul B. Larson - 2011 - Annals of the Japan Association for Philosophy of Science 19:57-86.
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  15.  14
    A Uniqueness Theorem for Iterations.Paul Larson - 2002 - Journal of Symbolic Logic 67 (4):1344-1350.
    If M is a countable transitive model of $ZFC+MA_{\aleph_{1}}$ , then for every real x there is a unique shortest iteration $j: M \rightarrow N$ with $x \in N$ , or none at all.
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  16.  31
    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.
  17.  26
    Itay Neeman. Games of Countable Length. Sets and Proofs , Edited by S. Barry Cooper and John K. Truss, London Mathematical Society Lecture Note Series, Vol. 258. Cambridge University Press, Cambridge, 1999, Pp. 159-196. - Itay Neeman. Unraveling Π11sets. Annals of Pure and Applied Logic, Vol. 106, No. 1–3 , Pp. 151-205. - Itay Neeman. Unraveling Π11sets, Revisited. Israel Journal of Mathematics, to Appear. [REVIEW]Paul B. Larson - 2005 - Bulletin of Symbolic Logic 11 (4):542-544.
  18.  50
    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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  19.  19
    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 Larson - 2002 - Bulletin of Symbolic Logic 8 (1):91-93.
  20. 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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  21. REVIEWS-Three Papers.I. Neeman & Paul B. Larson - 2005 - Bulletin of Symbolic Logic 11 (4):542-544.
     
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  22. REVIEWS-Two Papers-Proper Forcing and L (R).I. Neeman, J. Zapletal & Paul B. Larson - 2002 - Bulletin of Symbolic Logic 8 (4):548-549.
  23.  52
    Review: W. Hugh Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. [REVIEW]Paul B. Larson - 2002 - Bulletin of Symbolic Logic 8 (1):91-93.
  24.  36
    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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  25.  19
    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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  26.  24
    Games of Countable Length. Sets and Proofs.Paul B. Larson - 2005 - Bulletin of Symbolic Logic 11 (4):542-544.
  27.  5
    The Size of [Mathematical Formula].Paul Larson - 2000 - Archive for Mathematical Logic 39 (7):541-568.
  28.  23
    An $Mathbb{S}_{Max}$ Variation for One Souslin Tree.Paul Larson - 1999 - Journal of Symbolic Logic 64 (1):81-98.
    We present a variation of the forcing $\mathbb{S}_{max}$ as presented in Woodin [4]. Our forcing is a $\mathbb{P}_{max}$-style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree T$_G$ which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with T$_G$ being this minimal tree. In particular, in the extension this (...)
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  29.  12
    A Model of $$\mathsf {ZFA}+ \mathsf {PAC}$$ ZFA + PAC with No Outer Model of $$\mathsf {ZFAC}$$ ZFAC with the Same Pure Part.Paul Larson & Saharon Shelah - 2018 - Archive for Mathematical Logic 57 (7-8):853-859.
    We produce a model of \ such that no outer model of \ has the same pure sets, answering a question asked privately by Eric Hall.
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  30.  10
    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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  31.  8
    Splitting Stationary Sets From Weak Forms of Choice.Paul Larson & Saharon Shelah - 2009 - Mathematical Logic Quarterly 55 (3):299-306.
    Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular cardinals. This is a continuation of [4].
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  32.  15
    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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  33.  7
    Compact Spaces, Elementary Submodels, and the Countable Chain Condition.Lúcia R. Junqueira, Paul Larson & Franklin D. Tall - 2006 - Annals of Pure and Applied Logic 144 (1-3):107-116.
    Given a space in an elementary submodel M of H, define XM to be X∩M with the topology generated by . It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X=XM. Assuming in addition, the result holds for any compact XM satisfying the countable chain condition.
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  34.  19
    Proper Forcings and Absoluteness in LProper Forcing and L.Paul B. Larson, Itay Neeman & Jindrich Zapletal - 2002 - Bulletin of Symbolic Logic 8 (4):548.
  35.  12
    Martin's Maximum and The.Paul Larson - 2000 - Annals of Pure and Applied Logic 106 (1-3):135-149.
    Assuming the existence of a supercompact limit of supercompact cardinals, we modify the original consistency proof of Martin's Maximum to obtain a model in which MM holds but the axiom fails.
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