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  1. In Memoriam: James Earl Baumgartner.J. A. Larson - 2017 - Archive for Mathematical Logic 56 (7-8):877-909.
    James Earl Baumgartner came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory (...)
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  • Diamonds, Compactness, and Measure Sequences.Omer Ben-Neria - 2019 - Journal of Mathematical Logic 19 (1):1950002.
    We establish the consistency of the failure of the diamond principle on a cardinal κ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of κ in a Radin generic extension.
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  • -Sets of Reals.Haim Judah & Saharon Shelah - 1993 - Journal of Symbolic Logic 58 (1):72-80.
  • 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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  • Cardinal Preserving Ideals.Moti Gitik & Saharon Shelah - 1999 - Journal of Symbolic Logic 64 (4):1527-1551.
    We give some general criteria, when κ-complete forcing preserves largeness properties-like κ-presaturation of normal ideals on λ (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength "NS λ is ℵ 1 -preserving". for λ > ℵ 2.
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  • A Weak Variation of Shelah's I[Ω₂].William J. Mitchell - 2004 - Journal of Symbolic Logic 69 (1):94-100.
    We use a $\kappa^{+}-Mahlo$ cardinal to give a forcing construction of a model in which there is no sequence $\langle A_{\beta} : \beta \textless \omega_{2} \rangle$ of sets of cardinality $\omega_{1}$ such that $\{\lambda \textless \omega_{2} : \existsc \subset \lambda & (\bigcupc = \lambda otp(c) = \omega_{1} & \forall \beta \textless \lambda (c \cap \beta \in A_{\beta}))\}$ is stationary.
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  • Generic Σ3 1 Absoluteness.Sy D. Friedman - 2004 - Journal of Symbolic Logic 69 (1):73-80.
  • Full Reflection at a Measurable Cardinal.Thomas Jech & Jiří Witzany - 1994 - Journal of Symbolic Logic 59 (2):615-630.
    A stationary subset S of a regular uncountable cardinal κ reflects fully at regular cardinals if for every stationary set $T \subseteq \kappa$ of higher order consisting of regular cardinals there exists an α ∈ T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection (...)
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  • A Relative of the Approachability Ideal, Diamond and Non-Saturation.Assaf Rinot - 2010 - Journal of Symbolic Logic 75 (3):1035-1065.
    Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square _{\lambda}^{\ast}$ together with 2 λ = λ⁺ implies $\lozenge _{S}$ for every S ⊆ λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, (...)
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  • 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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  • Martin's Axiom and the Continuum.Haim Judah & Andrzej Rosłanowski - 1995 - Journal of Symbolic Logic 60 (2):374-391.
  • Infinitary Combinatorics and Modal Logic.Andreas Blass - 1990 - Journal of Symbolic Logic 55 (2):761-778.
    We show that the modal propositional logic G, originally introduced to describe the modality "it is provable that", is also sound for various interpretations using filters on ordinal numbers, for example the end-segment filters, the club filters, or the ineffable filters. We also prove that G is complete for the interpretation using end-segment filters. In the case of club filters, we show that G is complete if Jensen's principle □ κ holds for all $\kappa ; on the other hand, it (...)
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  • Martin's Axioms, Measurability and Equiconsistency Results.Jaime I. Ihoda & Saharon Shelah - 1989 - Journal of Symbolic Logic 54 (1):78-94.
    We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ (...)
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  • 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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  • 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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  • Summable Gaps.James Hirschorn - 2003 - Annals of Pure and Applied Logic 120 (1-3):1-63.
    It is proved, under Martin's Axiom, that all gaps in are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin's Axiom the Baire categoricity of all Δ31 non-Δ31-complete sets of reals requires a weakly compact cardinal.
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  • Simultaneous Stationary Reflection and Square Sequences.Yair Hayut & Chris Lambie-Hanson - 2017 - Journal of Mathematical Logic 17 (2):1750010.
    We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.
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  • A Characterization of {\ Square (\ Kappa^{+})} in Extender Models.Kyriakos Kypriotakis & Martin Zeman - 2013 - Archive for Mathematical Logic 52 (1-2):67-90.
    We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a ${\square(\kappa^{+})}$ -sequence if and only if there is a pair of stationary subsets of ${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$ without common reflection point of cofinality ${ < \kappa}$ which, in turn, is equivalent to the existence of a family of size ${ < \kappa}$ of stationary subsets of ${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$ without common reflection point of cofinality ${ < \kappa}$ . By (...)
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  • Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
  • Squares and Covering Matrices.Chris Lambie-Hanson - 2014 - Annals of Pure and Applied Logic 165 (2):673-694.
    Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof and subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between □κ and □ for a regular cardinal κ. We provide a detailed (...)
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  • Covering Theorems for the Core Model, and an Application to Stationary Set Reflection.Sean Cox - 2009 - Annals of Pure and Applied Logic 161 (1):66-93.
    We prove covering theorems for K, where K is the core model below the sharp for a strong cardinal, and give an application to stationary set reflection.
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  • 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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  • 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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  • Stationary Cardinals.Wenzhi Sun - 1993 - Archive for Mathematical Logic 32 (6):429-442.
    This paper will define a new cardinal called aStationary Cardinal. We will show that every weakly∏ 1 1 -indescribable cardinal is a stationary cardinal, every stationary cardinal is a greatly Mahlo cardinal and every stationary set of a stationary cardinal reflects. On the other hand, the existence of such a cardinal is independent of that of a∏ 1 1 -indescribable cardinal and the existence of a cardinal such that every stationary set reflects is also independent of that of a stationary (...)
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  • Exact Equiconsistency Results for Δ 3 1 -Sets of Reals.Haim Judah - 1992 - Archive for Mathematical Logic 32 (2):101-112.
    We improve a theorem of Raisonnier by showing that Cons(ZFC+every Σ 2 1 -set of reals in Lebesgue measurable+every Π 2 1 -set of reals isK σ-regular) implies Cons(ZFC+there exists an inaccessible cardinal). We construct, fromL, a model where every Δ 3 1 -sets of reals is Lebesgue measurable, has the property of Baire, and every Σ 2 1 -set of reals isK σ-regular. We prove that if there exists a Σ n+1 1 unbounded filter on ω, then there exists (...)
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  • The Eightfold Way.James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot & Dima Sinapova - 2018 - Journal of Symbolic Logic 83 (1):349-371.
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  • Sets of Reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
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  • 1\ Sets of Reals.J. Bagaria & W. H. Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.