Set Theory

Edited by Toby Meadows (University of Queensland)
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  1. Sobre os números transfinitos.Abdounur Oscar João, Junior Vecchio & Del Jacintho - 2013 - Scientiae Studia 11 (2):417-426.
    Este texto introduz a tradução do discurso de intitulado "Sobre os números transfinitos" ("Über transfinite Zahlen"), proferido por Henri Poincaré em 27 de abril de 1909, na Universidade de Göttingen. Após uma breve apresentação do pensamento do autor acerca dos fundamentos da aritmética, procura-se citar os aspectos mais relevantes da chamada crise dos fundamentos da matemática, para então introduzir a reformulação do conceito de predicatividade aventada no referido discurso sobre números transfinitos, contribuição compreendida como um recurso teórico necessário para a (...)
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  2. Some Results Concerning Strongly Compact Cardinals.Yoshihiro Abe - 1985 - Journal of Symbolic Logic 50 (4):874-880.
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  3. On the Existence of Universal Numberings for Finite Families of D.C.E. Sets.Kuanysh Abeshev - 2014 - Mathematical Logic Quarterly 60 (3):161-167.
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  4. Passages Between Finite and Infinite.Alexander Abian - 1978 - Notre Dame Journal of Formal Logic 19 (3):452-456.
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  5. The Consistency of the Continuum Hypothesis Via Synergistic Models.Alexander Abian - 1973 - Mathematical Logic Quarterly 19 (13):193-198.
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  6. On Definitions of Cuts and Completion of Partially Ordered Sets.Alexander Abian - 1968 - Mathematical Logic Quarterly 14 (19):299-302.
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  7. The Cardinality of Powersets in Finite Models of the Powerset Axiom.Alexander Abian & Wael A. Amin - 1991 - Notre Dame Journal of Formal Logic 32 (2):290-293.
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  8. On the Minimal Length of Sequences Representing Simply Ordered Sets.Alexander Abian & David Deever - 1967 - Mathematical Logic Quarterly 13 (1‐2):21-23.
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  9. On the Consistency and Independence of Some Set-Theoretical Axioms.Alexander Abian & Samuel LaMacchia - 1978 - Notre Dame Journal of Formal Logic 19 (1):155-158.
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  10. Some Consequences of the Axiom of Power-Set.Alexander Abian & Samuel Lamacchia - 1965 - Journal of Symbolic Logic 30 (3):293-294.
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  11. On the Intersection of Closed Unbounded Sets.U. Abraham & S. Shelah - 1986 - Journal of Symbolic Logic 51 (1):180-189.
    Forcing extensions yield models of ZFC in which a long sequence of club subsets of ω 1 has the following property: every subsequence of size ℵ 1 has a finite intersection.
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  12. 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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  13. Coding with Ladders a Well Ordering of the Reals.Uri Abraham & Saharon Shelah - 2002 - Journal of Symbolic Logic 67 (2):579-597.
    Any model of ZFC + GCH has a generic extension (made with a poset of size ℵ 2 ) in which the following hold: MA + 2 ℵ 0 = ℵ 2 +there exists a Δ 2 1 -well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on ω 1 . Therefore, the study of such ladders is a main concern of this article.
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  14. Martin's Axiom and Delta^2_1 Well-Ordering of the Reals.Uri Abraham & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):287-298.
    Assuming an inaccessible cardinal $\kappa$ , there is a generic extension in which $MA + 2^{\aleph_0} = \kappa$ holds and the reals have a $\Delta^2_1$ well-ordering.
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  15. Martin's Axiom and Well-Ordering of the Reals.Uri Abraham & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5):287-298.
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  16. Forcing Closed Unbounded Sets.Uri Abraham & Saharon Shelah - 1983 - Journal of Symbolic Logic 48 (3):643-657.
    We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.
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  17. Lévy Azriel. On Ackermann's Set Theory.W. Ackermann - 1960 - Journal of Symbolic Logic 25 (4):355.
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  18. Review: Azriel Levy, On Ackermann's Set Theory. [REVIEW]W. Ackermann - 1960 - Journal of Symbolic Logic 25 (4):355-355.
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  19. Mostowski Andrzej. Concerning a Problem of H. Scholz. Zeitschrift Für Mathematische Logik Und Grundlagen der Mathematik, Bd. 2 , S. 210–214. [REVIEW]W. Ackermann - 1959 - Journal of Symbolic Logic 24 (3):241.
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  20. Review: Andrzej Mostowski, Concerning a Problem of H. Scholz. [REVIEW]W. Ackermann - 1959 - Journal of Symbolic Logic 24 (3):241-241.
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  21. The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity.Amir D. Aczel - 2000 - Four Walls Eight Windows.
    From the end of the 19th century until his death, one of history's most brilliant mathematicians languished in an asylum. The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for his (...)
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  22. Aspects of General Topology in Constructive Set Theory.Peter Aczel - 2006 - Annals of Pure and Applied Logic 137 (1):3-29.
    Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures need (...)
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  23. Non-Well-Founded Sets.Peter Aczel - 1988 - Csli Lecture Notes.
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  24. Describing Ordinals Using Functionals of Transfinite Type.Peter Aczel - 1972 - Journal of Symbolic Logic 37 (1):35-47.
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  25. Binary Refinement Implies Discrete Exponentiation.Peter Aczel, Laura Crosilla, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2006 - Studia Logica 84 (3):361 - 368.
    Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary re.nement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary re.nement implies that the class of detachable subsets of a set form a set. Binary re.nement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was su.cient to prove that the (...)
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  26. Admissible Sets and the Saturation of Structures.Alan Adamson - 1978 - Annals of Mathematical Logic 14 (2):111-157.
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  27. The Theory of Models Proceedings of the 1963 International Symposium at Berkeley.J. W. Addison, Leon Henkin & Alfred Tarski - 1965
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  28. Ordinal Analysis and the Infinite Ramsey Theorem.Bahareh Afshari & Michael Rathjen - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 1--10.
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  29. Complete Axiomatisations of Properties of Finite Sets.Thomas Agotnes & Michal Walicki - 2008 - Logic Journal of the IGPL 16 (3):293-313.
    We study a logic whose formulae are interpreted as properties of a finite set over some universe. The language is propositional, with two unary operators inclusion and extension, both taking a finite set as argument. We present a basic Hilbert-style axiomatisation, and study its completeness. The main results are syntactic and semantic characterisations of complete extensions of the logic.
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  30. On Modal Μ-Calculus and Non-Well-Founded Set Theory.Luca Alberucci & Vincenzo Salipante - 2004 - Journal of Philosophical Logic 33 (4):343-360.
    A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.
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  31. On A Theory of Classes.C. Alkor - 1980 - Mathematical Logic Quarterly 26 (22‐24):337-342.
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  32. Notes on a New Definition of Infinite Cardinality.Allen D. Allen - 1976 - International Logic Review 7:57-60.
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  33. Connectedness and Compactness on Standard Sets.Ricardo Almeida - 2010 - Mathematical Logic Quarterly 56 (1):63-66.
    We present a nonstandard characterization of connected compact sets.
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  34. Recursively Enumerable Sets Which Are Uniform for Finite Extensions.Donald A. Alton - 1971 - Journal of Symbolic Logic 36 (2):271-287.
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  35. Automorphisms of Finite Order.D. A. Anapolitanos - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (33):565-575.
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  36. Partitions of Trees and {{Sf ACA}^Prime_{0}}.Bernard A. Anderson & Jeffry L. Hirst - 2009 - Archive for Mathematical Logic 48 (3-4):227-230.
    We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem ${{\sf ACA}^\prime_{0}}$ of reverse mathematics.
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  37. A Theory of Hyperfinite Sets.P. V. Andreev & E. I. Gordon - 2006 - Annals of Pure and Applied Logic 143 (1):3-19.
    We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS— which is based on the idea of the existence of proper subclasses of large finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS, and present some applications.
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  38. Finite Algebras of Relations Are Representable on Finite Sets.H. Andréka, I. Hodkinson & I. Németi - 1999 - Journal of Symbolic Logic 64 (1):243-267.
    Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.
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  39. Definable Open Sets As Finite Unions of Definable Open Cells.Simon Andrews - 2010 - Notre Dame Journal of Formal Logic 51 (2):247-251.
    We introduce CE- cell decomposition , a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure $\mathcal{R}$ admits CE-cell decomposition then any definable open set in $\mathcal{R}$ may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.
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  40. Letters: The Philosophy of Set Theory by Mary Tiles Oxford: Blackwell, 1989.Irving H. Anellis - 1993 - Philosophia Mathematica 1 (1):71-73.
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  41. Russell's Earliest Interpretation of Cantorian Set Theory, 1896–1900.Irving H. Anellis - 1987 - Philosophia Mathematica (1):1-31.
  42. Free Set Algebras Satisfying Systems of Equations.G. Aldo Antonelli - 1999 - Journal of Symbolic Logic 64 (4):1656-1674.
    In this paper we introduce the notion of a set algebra S satisfying a system E of equations. After defining a notion of freeness for such algebras, we show that, for any system E of equations, set algebras that are free in the class of structures satisfying E exist and are unique up to a bisimulation. Along the way, analogues of classical set-theoretic and algebraic properties are investigated.
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  43. Non-Well-Founded Sets Via Revision Rules.Gian Aldo Antonelli - 1994 - Journal of Philosophical Logic 23 (6):633 - 679.
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  44. There Exist Two Regressive Sets Whose Intersection is Not Regressive.K. I. Appel - 1967 - Journal of Symbolic Logic 32 (3):322-324.
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  45. A Stronger Definition of a Recursively Infinite Set.Charles H. Applebaum - 1973 - Notre Dame Journal of Formal Logic 14 (3):411-412.
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  46. Ad and Patterns of Singular Cardinals Below Θ.Arthur W. Apter - 1996 - Journal of Symbolic Logic 61 (1):225-235.
    Using Steel's recent result that assuming AD, in L[R] below Θ, κ is regular $\operatorname{iff} \kappa$ is measurable, we mimic below Θ certain earlier results of Gitik. In particular, we construct via forcing a model in which all uncountable cardinals below Θ are singular and a model in which the only regular uncountable cardinal below Θ is ℵ 1.
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  47. Successors of Singular Cardinals and Measurability Revisited.Arthur W. Apter - 1990 - Journal of Symbolic Logic 55 (2):492-501.
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  48. Blowing Up the Power Set of the Least Measurable.Arthur W. Apter & James Cummings - 2002 - Journal of Symbolic Logic 67 (3):915-923.
    We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of [1] by using the optimal hypothesis.
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  49. Identity Crises and Strong Compactness.Arthur W. Apter & James Cummings - 2000 - Journal of Symbolic Logic 65 (4):1895-1910.
    Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ 1 ,..., κ n are so that κ i for i = 1,..., n is both the i th measurable cardinal and κ + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.
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  50. The Consistency Strength of Choiceless Failures of SCH.Arthur W. Apter & Peter Koepke - 2010 - Journal of Symbolic Logic 75 (3):1066-1080.
    We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...)
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