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The iterative conception of set Review 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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An Order-Theoretic Account of Some Set-Theoretic Paradoxes with Thierry Libert Notre 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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Implementing Mathematical Objects in Set Theory Logique 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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Finite-to-one maps Journal 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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Sharvy’s Lucy and Benjamin Puzzle Studia Logica 90 (2). 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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Permutations and Wellfoundedness: The True Meaning of the Bizarre Arithmetic of Quine's NF Journal of Symbolic Logic 71 (1). 2006.
It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente
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Decidable Fragments of the Simple Theory of Types with Infinity and \$mathrm{NF}\$ with Anuj Dawar and Zachiri McKenzie Notre Dame Journal of Formal Logic 58 (3): 433-451. 2017.
We identify complete fragments of the simple theory of types with infinity and Quine’s new foundations set theory. We show that TSTI decides every sentence ϕ in the language of type theory that is in one of the following forms: ϕ=∀x1r1⋯∀xkrk∃y1s1⋯∃ylslθ where the superscripts denote the types of the variables, s1>⋯>sl, and θ is quantifier-free, ϕ=∀x1r1⋯∀xkrk∃y1s⋯∃ylsθ where the superscripts denote the types of the variables and θ is quantifier-free. This shows that NF decides every stratified se…Read more
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Sharvy’s Lucy and Benjamin Puzzle Studia 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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• 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.