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1A Consistent Higher‐Order Theory Without a (Higher‐Order) ModelMathematical Logic Quarterly 35 (5): 385-386. 2006.
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2Permutation Models in the Sense of Rieger‐BernaysMathematical Logic Quarterly 33 (3): 201-210. 2006.
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2Permutations and stratified formulae a preservation theoremMathematical Logic Quarterly 36 (5): 385-388. 2006.
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36Synonymy Questions Concerning the Quine SystemsJournal of Symbolic Logic 90 (4): 1779-1795. 2025.There are a variety of (“alternative”) axiomatic set theories available to mathematicians. It is worth asking how “alternative” they really are. Might they be no more than rephrasings of the theory (ZFC) that we already have? Here we give an account of the status of the Quine systems in this regard. Some are merely ZF in wolves’ clothing; some are genuine wolves.
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50Internal Automorphisms and Antimorphisms of Models of NfJournal of Symbolic Logic 90 (4): 1796-1800. 2025.It is shown that every model of NF admits a permutation model containing an internal automorphism.
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37Permutation Models in the Sense of Rieger‐BernaysMathematical Logic Quarterly 33 (3): 201-210. 1987.
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53Permutation Models in the Sense of Rieger-BernaysZeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (3): 201-210. 1987.
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48Reasoning 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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195Further consistency and independence results in NF obtained by the permutation methodJournal of Symbolic Logic 48 (2): 236-238. 1983.
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158Non-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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143The 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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174Term models for weak set theories with a universal setJournal of Symbolic Logic 52 (2): 374-387. 1987.
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133Ramsey’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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124A 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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383The iterative conception of setReview of Symbolic Logic 1 (1): 97-110. 2008.The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative conception of set is a good one, for al…Read more
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147Mathematical 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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147An 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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329ZF + "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 (Rieger-Bernays) permutation model of a model of ZF is a theorem of ZFB. (i.e.. ZFB is the theory of Rieger-Bernays permutation models of models of ZF) ZF and ZFAFA are both extensions of ZFB conservative for stratified formulæ. The class of models of ZFB is closed under creation of Rieger-Bernays permutation models
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42Permutations and stratified formulae a preservation theoremZeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (5): 385-388. 1990.
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207Finite-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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100Implementing Mathematical Objects in Set TheoryLogique Et Analyse 50 (197): 79-86. 2007.In general little thought is given to the general question of how to implement mathematical objects in set theory. It is clear that—at various times in the past—people have gone to considerable lengths to devise implementations with nice properties. There is a litera- ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that makes every set an ordered pair. The implementation of ordinals as Von Neumann ordinals i…Read more
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80Normal subgroups of infinite symmetric groups, with an application to stratified set theoryJournal of Symbolic Logic 74 (1): 17-26. 2009.
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83Erdös-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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117Sharvy’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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197End-extensions preserving power setJournal of Symbolic Logic 56 (1): 323-328. 1991.We consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a potentially useful new reduction of the consistency problem for Quine's NF, and a sharpening of another reduction of this problem due to Boffa.
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131Yablo's paradox and the omitting types theorem for propositional languagesLogique Et Analyse 54 (215): 323-326. 2011.
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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 |