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  1. Generic Vopěnka Cardinals and Models of ZF with Few $$\aleph _1$$ ℵ 1 -Suslin Sets.Trevor M. Wilson - 2019 - Archive for Mathematical Logic 58 (7-8):841-856.
    We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in models (...)
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  • Generic Vopěnka Cardinals and Models of ZF with Few $$\aleph _1$$ ℵ 1 -Suslin Sets.Trevor M. Wilson - 2019 - Archive for Mathematical Logic 58 (7-8):841-856.
    We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in models (...)
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  • Models as Fundamental Entities in Set Theory: A Naturalistic and Practice-Based Approach.Carolin Antos - forthcoming - Erkenntnis:1-28.
    This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to introduce models as (...)
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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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  • Combinatorics of Ultrafilters on Cohen and Random Algebras.Jörg Brendle & Francesco Parente - 2022 - Journal of Symbolic Logic 87 (1):109-126.
    We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.
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  • Equivalence Relations Which Are Borel Somewhere.William Chan - 2017 - Journal of Symbolic Logic 82 (3):893-930.
    The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence (...)
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  • Mathias and Set Theory.Akihiro Kanamori - 2016 - Mathematical Logic Quarterly 62 (3):278-294.
    On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.
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  • Hamel-Isomorphic Images of the Unit Ball.Jacek Cichoń & Przemysław Szczepaniak - 2010 - Mathematical Logic Quarterly 56 (6):625-630.
    In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ℝ2 and ℝ. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non-measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non-measurable.
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  • A Renaissance of Empiricism in the Recent Philosophy of Mathematics.Imre Lakatos - 1976 - British Journal for the Philosophy of Science 27 (3):201-223.
  • Mathematical Quantum Theory I: Random Ultrafilters as Hidden Variables.William Boos - 1996 - Synthese 107 (1):83 - 143.
    The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a (...)
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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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  • Forcing the Mapping Reflection Principle by Finite Approximations.Tadatoshi Miyamoto & Teruyuki Yorioka - 2021 - Archive for Mathematical Logic 60 (6):737-748.
    Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}. The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, (...)
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  • Distributive Proper Forcing Axiom and Cardinal Invariants.Huiling Zhu - 2013 - Archive for Mathematical Logic 52 (5-6):497-506.
    In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.
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  • Martin’s Conjecture and Strong Ergodicity.Simon Thomas - 2009 - Archive for Mathematical Logic 48 (8):749-759.
    In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.
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  • Forcing Axioms, Finite Conditions and Some More.Mirna Džamonja - 2013 - In Kamal Lodaya (ed.), Logic and its Applications. Springer. pp. 17--26.
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  • Inner Models From Extended Logics: Part 1.Juliette Kennedy, Menachem Magidor & Jouko Väänänen - 2020 - Journal of Mathematical Logic 21 (2):2150012.
    If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...
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  • Axioms of Symmetry: Throwing Darts at the Real Number Line.Chris Freiling - 1986 - Journal of Symbolic Logic 51 (1):190-200.
    We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...)
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  • BPFA and Projective Well-Orderings of the Reals.Andrés Eduardo Caicedo & Sy-David Friedman - 2011 - Journal of Symbolic Logic 76 (4):1126-1136.
    If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$ , then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface ${\mathrm{\Sigma }}_{4}^{1}$ , for many "consistently locally certified" relations R on $\mathrm{\mathbb{R}}$ . (...)
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  • On the Length of Borel Hierarchies.Arnorld W. Miller - 1979 - Annals of Mathematical Logic 16 (3):233.
  • Degrees of Functionals.Dag Normann - 1979 - Annals of Mathematical Logic 16 (3):269.
  • Second Order Arithmetic and Related Topics.K. R. Apt & W. Marek - 1974 - Annals of Mathematical Logic 6 (3):177.
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  • Long Projective Wellorderings.Leo Harrington - 1977 - Annals of Mathematical Logic 12 (1):1.
  • Happy Families.A. R. D. Mathias - 1977 - Annals of Mathematical Logic 12 (1):59.
  • Regularity Properties of Definable Sets of Reals.Jacques Stern - 1985 - Annals of Pure and Applied Logic 29 (3):289-324.
  • Models of Set Theory Containing Many Perfect Sets.John Truss - 1974 - Annals of Mathematical Logic 7 (2):197.
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  • Extensions of Kripke's Embedding Theorem.Jonathan Stavi - 1975 - Annals of Mathematical Logic 8 (4):345.
  • Madness in Vector Spaces.Iian B. Smythe - 2019 - Journal of Symbolic Logic 84 (4):1590-1611.
    We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability.
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  • A Basis Theorem for Perfect Sets.Marcia J. Groszek & Theodore A. Slaman - 1998 - Bulletin of Symbolic Logic 4 (2):204-209.
    We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair $M\subset N$ of models of set theory implying that every perfect set in N has an element in N which is not in M.
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  • Maximal Chains in the Turing Degrees.C. T. Chong & Liang Yu - 2007 - Journal of Symbolic Logic 72 (4):1219 - 1227.
    We study the problem of existence of maximal chains in the Turing degrees. We show that: 1. ZF+DC+"There exists no maximal chain in the Turing degrees" is equiconsistent with ZFC+"There exists an inaccessible cardinal"; 2. For all a ∈ 2ω.(ω₁)L[a] = ω₁ if and only if there exists a $\Pi _{1}^{1}[a]$ maximal chain in the Turing degrees. As a corollary, ZFC + "There exists an inaccessible cardinal" is equiconsistent with ZFC + "There is no (bold face) $\utilde{\Pi}{}_{1}^{1}$ maximal chain of (...)
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  • On the Structure of Δ 1 4 -Sets of Reals.Haim Judah & Otmar Spinas - 1995 - Archive for Mathematical Logic 34 (5):301-312.
    Assuming that an inaccessible cardinal exists, we construct a ZFC-model where every Δ 1 4 -set is measurable but there exists a Δ 1 4 -set without the property of Baire. By a result of Shelah, an inaccessible cardinal is necessary for this result.
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  • A Non-Implication Between Fragments of Martin’s Axiom Related to a Property Which Comes From Aronszajn Trees.Teruyuki Yorioka - 2010 - Annals of Pure and Applied Logic 161 (4):469-487.
    We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property . In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber 1 and for forcing notions with the (...)
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  • 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.
  • Δ< Sup> 1< Sub> 2-Sets of Reals.Jaime I. Ihoda & Saharon Shelah - 1989 - Annals of Pure and Applied Logic 42 (3):207-223.
  • Borel Sets and Ramsey's Theorem.Fred Galvin & Karel Prikry - 1973 - Journal of Symbolic Logic 38 (2):193-198.
  • On the Hanf Number of Souslin Logic.John P. Burgess - 1978 - Journal of Symbolic Logic 43 (3):568-571.
    We show it is consistent with ZFC that the Hanf number of Ellentuck's Souslin logic should be exactly $\beth_{\omega_2}$.
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  • Modest Theory of Short Chains. II.Yuri Gurevich & Saharon Shelah - 1979 - Journal of Symbolic Logic 44 (4):491-502.
    We analyse here the monadic theory of the rational order, the monadic theory of the real line with quantification over "small" subsets and models of these theories. We prove that the results are in some sense the best possible.
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  • Partitions and Filters.P. Matet - 1986 - Journal of Symbolic Logic 51 (1):12-21.
  • Some Filters of Partitions.Pierre Matet - 1988 - Journal of Symbolic Logic 53 (2):540-553.
  • Believing the Axioms. I.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):481-511.
  • Souslin Forcing.Jaime I. Ihoda & Saharon Shelah - 1988 - Journal of Symbolic Logic 53 (4):1188-1207.
    We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely MA(Γ + ℵ 0 ), and using the results on Souslin forcing we show that MA(Γ + ℵ 0 ) is consistent with the existence of a Souslin tree and with the splitting number s = ℵ 1 . We prove that MA(Γ + ℵ 0 ) proves the additivity of measure. Also we introduce (...)
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  • The Kunen-Miller Chart (Lebesgue Measure, the Baire Property, Laver Reals and Preservation Theorems for Forcing).Haim Judah & Saharon Shelah - 1990 - Journal of Symbolic Logic 55 (3):909-927.
    In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...)
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  • An Absoluteness Principle for Borel Sets.Greg Hjorth - 1998 - Journal of Symbolic Logic 63 (2):663-693.
  • Stationary Sets and Infinitary Logic.Saharon Shelah & Jouko Väänänen - 2000 - Journal of Symbolic Logic 65 (3):1311-1320.
    Let K 0 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ is disjoint from a club, and let K 1 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ contains a club. We prove that if $\lambda = \lambda^{ is regular, then no sentence of L λ+κ separates K 0 λ and K 1 λ . On the other hand, we prove that if $\lambda = \mu^+,\mu = \mu^{ , and a forcing axiom (...)
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  • Ladder Gaps Over Stationary Sets.Uri Abraham & Saharon Shelah - 2004 - Journal of Symbolic Logic 69 (2):518-532.
    For a stationary set S⊆ ω1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over ω1∖ S there exists a gap with no subgap that is E-Hausdorff.A new type of chain condition, called polarized chain condition, is introduced. (...)
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  • Ladder Gaps Over Stationary Sets.Uri Abraham & Saharon Shelah - 2004 - Journal of Symbolic Logic 69 (2):518 - 532.
    For a stationary set $S \subseteq \omega_{1}$ and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over \omega_{1} \ S$ there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, called polarized chain (...)
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  • Projective Well-Orderings and Bounded Forcing Axioms.Andrés Eduardo Caicedo - 2005 - Journal of Symbolic Logic 70 (2):557 - 572.
    In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.
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  • Projective Well-Orderings and Bounded Forcing Axioms.Andrés Eduardo Caicedo - 2005 - Journal of Symbolic Logic 70 (2):557-572.
    In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.
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  • Complexity of Reals in Inner Models of Set Theory.Boban Velickovic & W. Hugh Woodin - 1998 - Annals of Pure and Applied Logic 92 (3):283-295.
    We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set (...)
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  • Universal Sets for Pointsets Properly on the N Th Level of the Projective Hierarchy.Greg Hjorth, Leigh Humphries & Arnold W. Miller - 2013 - Journal of Symbolic Logic 78 (1):237-244.
    The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.
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  • Weak Distributivity Implying Distributivity.Dan Hathaway - 2016 - Journal of Symbolic Logic 81 (2):711-717.
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