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PFA Implies ADL(R)

Journal of Symbolic Logic 70 (4):1255 - 1296 (2005)

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  1. Two Upper Bounds on Consistency Strength of $Negsquare{Aleph_{Omega}}$ and Stationary Set Reflection at Two Successive $Aleph{N}$.Martin Zeman - 2017 - Notre Dame Journal of Formal Logic 58 (3):409-432.
    We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...)
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  • Definable MAD Families and Forcing Axioms.Vera Fischer, David Schrittesser & Thilo Weinert - 2021 - Annals of Pure and Applied Logic 172 (5):102909.
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  • Determinacy From Strong Compactness of Ω1.Nam Trang & Trevor M. Wilson - 2021 - Annals of Pure and Applied Logic 172 (6):102944.
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  • Tame Failures of the Unique Branch Hypothesis and Models of ADℝ + Θ is Regular.Grigor Sargsyan & Nam Trang - 2016 - Journal of Mathematical Logic 16 (2):1650007.
    In this paper, we show that the failure of the unique branch hypothesis for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66 903–923; Core models with more Woodin cardinals, J. Symbolic Logic (...)
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  • Combinatorial Dichotomies in Set Theory.Stevo Todorcevic - 2011 - Bulletin of Symbolic Logic 17 (1):1-72.
    We give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further (...)
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  • Scales in K at the End of a Weak Gap.J. R. Steel - 2008 - Journal of Symbolic Logic 73 (2):369-390.
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  • Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  • Rado's Conjecture Implies That All Stationary Set Preserving Forcings Are Semiproper.Philipp Doebler - 2013 - Journal of Mathematical Logic 13 (1):1350001.
    Todorčević showed that Rado's Conjecture implies CC*, a strengthening of Chang's Conjecture. We generalize this by showing that also CC**, a global version of CC*, follows from RC. As a corollary we obtain that RC implies Semistationary Reflection and, i.e. the statement that all forcings that preserve the stationarity of subsets of ω1 are semiproper.
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  • Determinacy in L.Nam Trang - 2014 - Journal of Mathematical Logic 14 (1):1450006.
    Assume V = L ⊨ ZF + DC + Θ > ω2 + μ is a normal fine measure on.
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  • Nontame Mouse From the Failure of Square at a Singular Strong Limit Cardinal.Grigor Sargsyan - 2014 - Journal of Mathematical Logic 14 (1):1450003.
    Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216 89–117] and Steel [PFA implies AD L, J. Symbolic Logic70 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong (...)
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  • 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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  • Descriptive Inner Model Theory.Grigor Sargsyan - 2013 - Bulletin of Symbolic Logic 19 (1):1-55.
    The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One particular motivation for resolving MSC is that (...)
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  • The Envelope of a Pointclass Under a Local Determinacy Hypothesis.Trevor M. Wilson - 2015 - Annals of Pure and Applied Logic 166 (10):991-1018.
  • The Strength of Choiceless Patterns of Singular and Weakly Compact Cardinals.Daniel Busche & Ralf Schindler - 2009 - Annals of Pure and Applied Logic 159 (1-2):198-248.
    We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that , the Axiom of Determinacy, holds in the of a generic extension of : every uncountable cardinal is singular, and every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.
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  • Characterizing All Models in Infinite Cardinalities.Lauri Keskinen - 2013 - Annals of Pure and Applied Logic 164 (3):230-250.
    Fix a cardinal κ. We can ask the question: what kind of a logic L is needed to characterize all models of cardinality κ up to isomorphism by their L-theories? In other words: for which logics L it is true that if any models A and B of cardinality κ satisfy the same L-theory then they are isomorphic?It is always possible to characterize models of cardinality κ by their Lκ+,κ+-theories, but we are interested in finding a “small” logic L, i.e., (...)
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  • Proper and Piecewise Proper Families of Reals.Victoria Gitman - 2009 - Mathematical Logic Quarterly 55 (5):542-550.
    I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω1.Here, (...)
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