26 found
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  1.  37
    Large Cardinals Beyond Choice.Joan Bagaria, Peter Koellner & W. Hugh Woodin - 2019 - Bulletin of Symbolic Logic 25 (3):283-318.
    The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...)
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  2.  20
    Bounded Forcing Axioms as Principles of Generic Absoluteness.Joan Bagaria - 2000 - Archive for Mathematical Logic 39 (6):393-401.
    We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then (...)
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  3.  66
    C (N)-Cardinals.Joan Bagaria - 2012 - Archive for Mathematical Logic 51 (3-4):213-240.
    For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such that V α is a Σ n elementary substructure of V. We say that κ is a C (n) -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C (n). By analyzing the notion of C (n)-cardinal at various levels of the usual hierarchy of large cardinal principles we show (...)
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  4. $\Underset{\Tilde}{\Delta}^1_n$ Sets Of Reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
  5.  12
    Superstrong and Other Large Cardinals Are Never Laver Indestructible.Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba - 2016 - Archive for Mathematical Logic 55 (1-2):19-35.
    Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...)
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  6. Solovay Models and Forcing Extensions.Joan Bagaria & Roger Bosch - 2004 - Journal of Symbolic Logic 69 (3):742-766.
    We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly-̰Σ₃¹ absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for Σ₃¹ absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact (...)
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  7.  14
    On Colimits and Elementary Embeddings.Joan Bagaria & Andrew Brooke-Taylor - 2013 - Journal of Symbolic Logic 78 (2):562-578.
    We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as $\alpha$-strongly compact and $C^{(n)}$-extendible cardinals.
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  8.  13
    Fragments of Martin's Axiom and Δ13 Sets of Reals.Joan Bagaria - 1994 - Annals of Pure and Applied Logic 69 (1):1-25.
    We strengthen a result of Harrington and Shelah by showing that, unless ω1 is an inaccessible cardinal in L, a relatively weak fragment of Martin's axiom implies that there exists a δ13 set of reals without the property of Baire.
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  9.  14
    Parameterized Partition Relations on the Real Numbers.Joan Bagaria & Carlos A. Di Prisco - 2009 - Archive for Mathematical Logic 48 (2):201-226.
    We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition (...)
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  10.  12
    Generic Absoluteness.Joan Bagaria & Sy D. Friedman - 2001 - Annals of Pure and Applied Logic 108 (1-3):3-13.
    We explore the consistency strength of Σ 3 1 and Σ 4 1 absoluteness, for a variety of forcing notions.
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  11.  18
    Proper Forcing Extensions and Solovay Models.Joan Bagaria & Roger Bosch - 2003 - Archive for Mathematical Logic 43 (6):739-750.
    We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the consistency strength of the absoluteness of under (...)
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  12.  40
    A Characterization of Martin's Axiom in Terms of Absoluteness.Joan Bagaria - 1997 - Journal of Symbolic Logic 62 (2):366-372.
    Martin's axiom is equivalent to the statement that the universe is absolute under ccc forcing extensions for Σ 1 sentences with a subset of $\kappa, \kappa , as a parameter.
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  13.  21
    Projective Forcing.Joan Bagaria & Roger Bosch - 1997 - Annals of Pure and Applied Logic 86 (3):237-266.
    We study the projective posets and their properties as forcing notions. We also define Martin's axiom restricted to projective sets, MA, and show that this axiom is weaker than full Martin's axiom by proving the consistency of ZFC + ¬lCH + MA with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists κ < 20 such that 20 < 2k”.
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  14.  7
    On Coding Uncountable Sets by Reals.Joan Bagaria & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.
    If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y (...)
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  15.  19
    Sets of Reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
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  16.  5
    Fragments of Martin's Axiom and Δ< Sup> 1< Sub> 3 Sets of Reals.Joan Bagaria - 1994 - Annals of Pure and Applied Logic 69 (1):1-25.
  17.  11
    The Consistency Strength of Hyperstationarity.Joan Bagaria, Menachem Magidor & Salvador Mancilla - 2019 - Journal of Mathematical Logic 20 (1):2050004.
    We introduce the large-cardinal notions of ξ-greatly-Mahlo and ξ-reflection cardinals and prove (1) in the constructible universe, L, the first ξ-reflection cardinal, for ξ a successor ordinal, is strictly between the first ξ-greatly-Mahlo and the first Π1ξ-indescribable cardinals, (2) assuming the existence of a ξ-reflection cardinal κ in L, ξ a successor ordinal, there exists a forcing notion in L that preserves cardinals and forces that κ is (ξ+1)-stationary, which implies that the consistency strength of the existence of a (ξ+1)-stationary (...)
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  18.  9
    Bounded Forcing Axioms and the Continuum.David Asperó & Joan Bagaria - 2001 - Annals of Pure and Applied Logic 109 (3):179-203.
    We show that bounded forcing axioms are consistent with the existence of -gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...)
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  19.  13
    Saharon Shelah and Hugh Woodin. Large Cardinals Imply That Every Reasonably Definable Set of Reals is Lebesgue Measurable. Israel Journal of Mathematics, Vol. 70 , Pp. 381–394. [REVIEW]Joan Bagaria - 2002 - Bulletin of Symbolic Logic 8 (4):543-545.
  20.  23
    On ${\Omega _1}$-Strongly Compact Cardinals.Joan Bagaria & Menachem Magidor - 2014 - Journal of Symbolic Logic 79 (1):266-278.
  21.  33
    Review: Saharon Shelah, Hugh Woodin, Large Cardinals Imply That Every Reasonably Definable Set of Reals Is Lebesgue Measurable. [REVIEW]Joan Bagaria - 2002 - Bulletin of Symbolic Logic 8 (4):543-545.
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  22.  12
    Preface.Klaus Ambos-Spies, Joan Bagaria, Enrique Casanovas & Ulrich Kohlenbach - 2013 - Annals of Pure and Applied Logic 164 (12):1177.
  23.  9
    Preface.Joan Bagaria, Yiannis Moschovakis, Margarita Otero & Ivan Soskov - 2011 - Annals of Pure and Applied Logic 162 (7):489.
  24.  6
    On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals.Joan Bagaria & Jouko Väänänen - 2016 - Journal of Symbolic Logic 81 (2):584-604.
  25.  5
    On Turing’s Legacy in Mathematical Logic and the Foundations of Mathematics.Joan Bagaria - 2013 - Arbor 189 (764):a079.
  26. Set Theory: Techniques and Applications.Carlos Augusto Di Prisco, Jean A. Larson, Joan Bagaria & A. R. D. Mathias - 2000 - Studia Logica 66 (3):426-428.