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Set Theory

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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. Alexander Abian (1978). Passages Between Finite and Infinite. Notre Dame Journal of Formal Logic 19 (3):452-456.
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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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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  10. 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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  11. 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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  12. Peter Aczel (1972). Describing Ordinals Using Functionals of Transfinite Type. Journal of Symbolic Logic 37 (1):35-47.
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  13. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. Irving H. Anellis (1987). Russell's Earliest Interpretation of Cantorian Set Theory, 1896–1900. Philosophia Mathematica (1):1-31.
  23. 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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  24. Gian Aldo Antonelli (1994). Non-Well-Founded Sets Via Revision Rules. Journal of Philosophical Logic 23 (6):633 - 679.
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  25. 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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  26. 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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  27. 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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  28. Arthur W. Apter (1990). Successors of Singular Cardinals and Measurability Revisited. Journal of Symbolic Logic 55 (2):492-501.
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  29. 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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  30. 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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  31. 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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  32. F. G. Asenjo (1970). Generalized Reals. Notre Dame Journal of Formal Logic 11 (4):473-476.
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  33. 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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  34. F. G. Asenjo (1963). Relations Irreducible to Classes. Notre Dame Journal of Formal Logic 4 (3):193-200.
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  35. 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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  36. 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.
  37. 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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  38. A. Avron & B. Konikowska (2008). Rough Sets and 3-Valued Logics. Studia Logica 90 (1):69 - 92.
    In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is (...)
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  39. Arnon Avron, A New Approach to Predicative Set Theory.
    We suggest a new framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically (...)
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  40. Arnon Avron, Constructibility and Decidability Versus Domain Independence and Absoluteness.
    We develop a unified framework for dealing with constructibility and absoluteness in set theory, decidability of relations in effective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domain-independence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability.
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  41. S. Awodey, N. Gambino & M. A. Warren (2009). Lawvere-Tierney Sheaves in Algebraic Set Theory. Journal of Symbolic Logic 74 (3):861 - 890.
    We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.
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  42. Steve Awodey (2009). From Sets to Types to Categories to Sets. Philosophical Explorations.
    Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...)
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  43. Steve Awodey (2008). A Brief Introduction to Algebraic Set Theory. Bulletin of Symbolic Logic 14 (3):281-298.
    This brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of (...)
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  44. Steve Awodey, Carsten Butz & Alex Simpson (2007). Relating First-Order Set Theories and Elementary Toposes. Bulletin of Symbolic Logic 13 (3):340-358.
    We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...)
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  45. Steve Awodey & Henrik Forssell, Algebraic Models of Intuitionistic Theories of Sets and Classes.
    This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in [2] by introducing a (...)
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  46. N. D. B. (1960). Axiomatic Set Theory. Review of Metaphysics 14 (1):175-175.
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  47. Serikzhan A. Badaev & Steffen Lempp (2009). A Decomposition of the Rogers Semilattice of a Family of D.C.E. Sets. Journal of Symbolic Logic 74 (2):618-640.
    Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up (...)
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  48. Joan Bagaria & W. Hugh Woodin (1997). $\Underset{\Tilde}{\Delta}^1_n$ Sets of Reals. Journal of Symbolic Logic 62 (4):1379-1428.
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  49. Sidney C. Bailin (1988). A Normalization Theorem for Set Theory. Journal of Symbolic Logic 53 (3):673-695.
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  50. Bektur Baizhanov, John T. Baldwin & Saharon Shelah (2005). Subsets of Superstable Structures Are Weakly Benign. Journal of Symbolic Logic 70 (1):142 - 150.
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