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5Alternative Set TheoriesIn Dov Gabbay (ed.), The Handbook of the History of Logic, Elsevier. 2009.
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1Permutation Models in the Sense of Rieger‐BernaysMathematical Logic Quarterly 33 (3): 201-210. 1987.
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1Permutation Models in the Sense of Rieger-BernaysZeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (3): 201-210. 1987.
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4Non-well-foundedness of well-orderable power setsJournal of Symbolic Logic 68 (3): 879-884. 2003.Tarski [5] showed that for any setX, its setω(X) of well-orderable subsets has cardinality strictly greater than that ofX, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation ∣ω(X)∣ = ∣Y∣. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation.
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3Reasoning About Theoretical EntitiesWorld Scientific. 2003.As such this book fills a void in the philosophical literature and presents a challenge to every would-be (anti-)reductionist.
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8Ramsey’s theorem and König’s LemmaArchive for Mathematical Logic 46 (1): 37-42. 2007.We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.
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21Non-well-foundedness of well-orderable power setsJournal of Symbolic Logic 68 (3): 879-884. 2003.Tarski [5] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w (X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation |P(X)| = |Y|
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7The status of the axiom of choice in set theory with a universal setJournal of Symbolic Logic 50 (3): 701-707. 1985.
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2Ramsey’s theorem and König’s LemmaArchive for Mathematical Logic 46 (1): 37-42. 2007.We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice
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9Term models for weak set theories with a universal setJournal of Symbolic Logic 52 (2): 374-387. 1987.
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16Further consistency and independence results in NF obtained by the permutation methodJournal of Symbolic Logic 48 (2): 236-238. 1983.
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17A Note on Freedom from Detachment in the Logic of ParadoxNotre Dame Journal of Formal Logic 54 (1): 15-20. 2013.We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing
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Zf + “every Set Is The Same Size As A Wellfounded Set”Journal of Symbolic Logic 68 (1): 1-4. 2003.Let ZFB be ZF + “every set is the same size as a wellfounded set”. Then the following are true.Every sentence true in every permutation model of a model of ZF is a theorem of ZFB. ZF and ZFAFA are both extensions of ZFB conservative for stratified formul{\ae}.{The class of models of ZFB is closed under creation of Rieger-Bernays permutation models.
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5Sharvy’s Lucy and Benjamin PuzzleStudia Logica 90 (2): 249-256. 2008.Sharvy’s puzzle concerns a situation in which common knowledge of two parties is obtained by repeated observation each of the other, no fixed point being reached in finite time. Can a fixed point be reached?
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2Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NFPhilosophia Mathematica 24 (1). 2016.Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not…Read more
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6An Order-Theoretic Account of Some Set-Theoretic ParadoxesNotre Dame Journal of Formal Logic 52 (1): 1-19. 2011.We present an order-theoretic analysis of set-theoretic paradoxes. This analysis will show that a large variety of purely set-theoretic paradoxes (including the various Russell paradoxes as well as all the familiar implementations of the paradoxes of Mirimanoff and Burali-Forti) are all instances of a single limitative phenomenon
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2Yablo's Paradox and the Omitting Types Theorem for Propositional LanguagesLogique Et Analyse 54 (215): 323. 2011.
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6Permutations and stratified formulae a preservation theoremMathematical Logic Quarterly 36 (5): 385-388. 1990.
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16Finite-to-one mapsJournal of Symbolic Logic 68 (4): 1251-1253. 2003.It is shown in ZF (without choice) that if there is a finite-to-one map P(X) → X, then X is finite
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2Erdös-Rado without ChoiceJournal of Symbolic Logic 72 (3). 2007.A version of the Erdös-Rado theorem on partitions of the unordered n-tuples from uncountable sets is proved, without using the axiom of choice. The case with exponent 1 is just the Sierpinski-Hartogs' result that $\aleph (\alpha)\leq 2^{2^{2^{\alpha}}}$
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3Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NFPhilosophia Mathematica 24 (1): 50-59. 2016.Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted _aussonderung_ but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for "low" sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is n…Read more
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2Normal subgroups of infinite symmetric groups, with an application to stratified set theoryJournal of Symbolic Logic 74 (1): 17-26. 2009.
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5Permutations and stratified formulae a preservation theoremZeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (5): 385-388. 1990.
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Cambridge UniversityRetired faculty
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Cambridge UniversityRetired faculty
Cambridge, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
Science, Logic, and Mathematics |
Areas of Interest
Science, Logic, and Mathematics |