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Set Theory

Edited by Toby Meadows (University of Aberdeen)
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  1. Yoshihiro Abe (1985). Some Results Concerning Strongly Compact Cardinals. Journal of Symbolic Logic 50 (4):874-880.
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  2. Kuanysh Abeshev (forthcoming). On the Existence of Universal Numberings for Finite Families of D.C.E. Sets. Mathematical Logic Quarterly:n/a-n/a.
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  3. Alexander Abian (1978). Passages Between Finite and Infinite. Notre Dame Journal of Formal Logic 19 (3):452-456.
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  4. Alexander Abian (1973). The Consistency of the Continuum Hypothesis Via Synergistic Models. Mathematical Logic Quarterly 19 (13):193-198.
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  5. Alexander Abian (1968). On Definitions of Cuts and Completion of Partially Ordered Sets. Mathematical Logic Quarterly 14 (19):299-302.
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  6. Alexander Abian & Wael A. Amin (1991). The Cardinality of Powersets in Finite Models of the Powerset Axiom. Notre Dame Journal of Formal Logic 32 (2):290-293.
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  7. Alexander Abian & David Deever (1967). On the Minimal Length of Sequences Representing Simply Ordered Sets. Mathematical Logic Quarterly 13 (1‐2):21-23.
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  8. Alexander Abian & Samuel LaMacchia (1978). On the Consistency and Independence of Some Set-Theoretical Axioms. Notre Dame Journal of Formal Logic 19 (1):155-158.
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  9. Alexander Abian & Samuel Lamacchia (1965). Some Consequences of the Axiom of Power-Set. Journal of Symbolic Logic 30 (3):293-294.
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  10. U. Abraham & S. Shelah (1986). On the Intersection of Closed Unbounded Sets. 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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  11. Uri Abraham & Saharon Shelah (2004). Ladder Gaps Over Stationary Sets. 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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  12. Uri Abraham & Saharon Shelah (2002). Coding with Ladders a Well Ordering of the Reals. 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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  13. Uri Abraham & Saharon Shelah (1996). Martin's Axiom and Delta^2_1 Well-Ordering of the Reals. 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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  14. Uri Abraham & Saharon Shelah (1996). Martin's Axiom and Well-Ordering of the Reals. Archive for Mathematical Logic 35 (5):287-298.
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  15. Uri Abraham & Saharon Shelah (1983). Forcing Closed Unbounded Sets. 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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  16. Amir D. Aczel (2000). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. 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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  17. Peter Aczel (2006). Aspects of General Topology in Constructive Set Theory. 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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  18. Peter Aczel (1988). Non-Well-Founded Sets. Csli Lecture Notes.
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  19. Peter Aczel (1972). Describing Ordinals Using Functionals of Transfinite Type. Journal of Symbolic Logic 37 (1):35-47.
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  20. Peter Aczel, Laura Crosilla, Hajime Ishihara, Erik Palmgren & Peter Schuster (2006). Binary Refinement Implies Discrete Exponentiation. 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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  21. Alan Adamson (1978). Admissible Sets and the Saturation of Structures. Annals of Mathematical Logic 14 (2):111-157.
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  22. J. W. Addison, Leon Henkin & Alfred Tarski (1965). The Theory of Models Proceedings of the 1963 International Symposium at Berkeley.
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  23. Bahareh Afshari & Michael Rathjen (2012). Ordinal Analysis and the Infinite Ramsey Theorem. In S. Barry Cooper (ed.), How the World Computes. 1--10.
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  24. Thomas Agotnes & Michal Walicki (2008). Complete Axiomatisations of Properties of Finite Sets. 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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  25. Luca Alberucci & Vincenzo Salipante (2004). On Modal Μ-Calculus and Non-Well-Founded Set Theory. 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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  26. C. Alkor (1980). On A Theory of Classes. Mathematical Logic Quarterly 26 (22‐24):337-342.
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  27. Allen D. Allen (1976). Notes on a New Definition of Infinite Cardinality. International Logic Review 7:57-60.
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  28. Ricardo Almeida (2010). Connectedness and Compactness on Standard Sets. Mathematical Logic Quarterly 56 (1):63-66.
    We present a nonstandard characterization of connected compact sets.
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  29. Donald A. Alton (1971). Recursively Enumerable Sets Which Are Uniform for Finite Extensions. Journal of Symbolic Logic 36 (2):271-287.
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  30. Bernard A. Anderson & Jeffry L. Hirst (2009). Partitions of Trees and {{Sf ACA}^Prime_{0}}. 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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  31. P. V. Andreev & E. I. Gordon (2006). A Theory of Hyperfinite Sets. 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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  32. H. Andréka, I. Hodkinson & I. Németi (1999). Finite Algebras of Relations Are Representable on Finite Sets. 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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  33. Simon Andrews (2010). Definable Open Sets As Finite Unions of Definable Open Cells. 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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  34. Irving H. Anellis (1993). Letters: The Philosophy of Set Theory by Mary Tiles Oxford: Blackwell, 1989. Philosophia Mathematica 1 (1):71-73.
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  35. Irving H. Anellis (1987). Russell's Earliest Interpretation of Cantorian Set Theory, 1896–1900. Philosophia Mathematica (1):1-31.
  36. G. Aldo Antonelli (1999). Free Set Algebras Satisfying Systems of Equations. 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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  37. Gian Aldo Antonelli (1994). Non-Well-Founded Sets Via Revision Rules. Journal of Philosophical Logic 23 (6):633 - 679.
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  38. K. I. Appel (1967). There Exist Two Regressive Sets Whose Intersection is Not Regressive. Journal of Symbolic Logic 32 (3):322-324.
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  39. Charles H. Applebaum (1973). A Stronger Definition of a Recursively Infinite Set. Notre Dame Journal of Formal Logic 14 (3):411-412.
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  40. Arthur W. Apter (1996). Ad and Patterns of Singular Cardinals Below Θ. 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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  41. Arthur W. Apter (1990). Successors of Singular Cardinals and Measurability Revisited. Journal of Symbolic Logic 55 (2):492-501.
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  42. Arthur W. Apter & James Cummings (2002). Blowing Up the Power Set of the Least Measurable. 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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  43. Arthur W. Apter & James Cummings (2000). Identity Crises and Strong Compactness. 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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  44. Arthur W. Apter & Peter Koepke (2010). The Consistency Strength of Choiceless Failures of SCH. 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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  45. F. G. Asenjo (1970). Generalized Reals. Notre Dame Journal of Formal Logic 11 (4):473-476.
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  46. F. G. Asenjo (1967). Rings of Term-Relation Numbers as Non-Standard Models. Notre Dame Journal of Formal Logic 8 (1-2):24-26.
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  47. F. G. Asenjo (1963). Relations Irreducible to Classes. Notre Dame Journal of Formal Logic 4 (3):193-200.
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  48. David Asperó, John Krueger & Yasuo Yoshinobu (2009). Dense Non-Reflection for Stationary Collections of Countable Sets. Annals of Pure and Applied Logic 161 (1):94-108.
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  49. Jeremy Avigad (2010). Proof Theory. Gödel and the Metamathematical Tradition. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic
  50. Jeremy Avigad (2000). Interpreting Classical Theories in Constructive Ones. Journal of Symbolic Logic 65 (4):1785-1812.
    A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.
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