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48Fragile measurabilityJournal of Symbolic Logic 59 (1): 262-282. 1994.Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of κ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves $\kappa^{
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307The set-theoretic multiverseReview of Symbolic Logic 5 (3): 416-449. 2012.The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic …Read more
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97A simple maximality principleJournal of Symbolic Logic 68 (2): 527-550. 2003.In this paper, following an idea of Christophe Chalons. I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence varphi holding in some forcing extension $V^P$ and all subsequent extensions $V^{P\ast Q}$ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\lozenge \square \varphi) \Rightarrow \square \varphi$ , and is equ…Read more
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78Diamond (on the regulars) can fail at any strongly unfoldable cardinalAnnals of Pure and Applied Logic 144 (1-3): 83-95. 2006.If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which κ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser
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25Superdestructibility: A Dual to Laver's IndestructibilityJournal of Symbolic Logic 63 (2): 549-554. 1998.After small forcing, any $ -closed forcing will destroy the supercompactness and even the strong compactness of κ
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83Pointwise definable models of set theoryJournal of Symbolic Logic 78 (1): 139-156. 2013.A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is po…Read more
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166Is the Dream Solution of the Continuum Hypothesis Attainable?Notre Dame Journal of Formal Logic 56 (1): 135-145. 2015.The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable
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105With infinite utility, more needn't be betterAustralasian Journal of Philosophy 78 (2). 2000.This Article does not have an abstract
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40Changing the heights of automorphism towersAnnals of Pure and Applied Logic 102 (1-2): 139-157. 2000.If G is a centreless group, then τ denotes the height of the automorphism tower of G. We prove that it is consistent that for every cardinal λ and every ordinal α
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37What is the theory without power set?Mathematical Logic Quarterly 62 (4-5): 391-406. 2016.We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there a…Read more
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21The halting problem is almost always decidableNotre Dame Journal of Formal Logic 47 (4): 515-524. 2006.
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53Large cardinals need not be large in HODAnnals of Pure and Applied Logic 166 (11): 1186-1198. 2015.
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36Post’s Problem for ordinal register machines: An explicit approachAnnals of Pure and Applied Logic 160 (3): 302-309. 2009.We provide a positive solution for Post’s Problem for ordinal register machines, and also prove that these machines and ordinal Turing machines compute precisely the same partial functions on ordinals. To do so, we construct ordinal register machine programs which compute the necessary functions. In addition, we show that any set of ordinals solving Post’s Problem must be unbounded in the writable ordinals
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22Infinite time Turing machinesJournal of Symbolic Logic 65 (2): 567-604. 2000.We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Everyset. for example, is decidable by such machines, and the semi-decidable sets form a portion of thesets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.
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228Gap forcing: Generalizing the lévy-Solovay theoremBulletin of Symbolic Logic 5 (2): 264-272. 1999.The Lévy-Solovay Theorem [8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardi…Read more
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58The Wholeness Axioms and V=HODArchive for Mathematical Logic 40 (1): 1-8. 2001.If the Wholeness Axiom wa $_0$ is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa $_0$ is finitely axiomatizable
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5A Simple Maximality PrincipleJournal of Symbolic Logic 68 (2): 527-550. 2003.In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension $V\P$ and all subsequent extensions V\P*\Qdot holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $\implies\necessaryφ$, and is equivalent to the modal theory S5. In this artic…Read more
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75New inconsistencies in infinite utilitarianism: Is every world good, bad or neutral?Australasian Journal of Philosophy 80 (2). 2002.In the context of worlds with infinitely many bearers of utility, we argue that several collections of natural Utilitarian principles--principles which are certainly true in the classical finite Utilitarian context and which any Utilitarian would find appealing--are inconsistent.
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58Small forcing makes any cardinal superdestructibleJournal of Symbolic Logic 63 (1): 51-58. 1998.Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal κ becomes superdestructible--any further
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97Exactly controlling the non-supercompact strongly compact cardinalsJournal of Symbolic Logic 68 (2): 669-688. 2003.We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set o…Read more
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P f_≠ NP _f_ for almost all _fMathematical Logic Quarterly 49 (5): 536-540. 2003.We discuss the question of Ralf‐Dieter Schindler whether for infinite time Turing machines Pf = NPf can be true for any function f from the reals into ω1. We show that “almost everywhere” the answer is negative.
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University of OxfordFaculty of Philosophy, University CollegeProfessor of Logic, Sir Peter Strawson Fellow
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Oxford, England, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
3 more
Mathematical Logic |
The Infinite |
Logic and Philosophy of Logic |
Set Theory |
Philosophy of Mathematics |
Hypercomputation |
Theory of Computation |
Modal Logic |
Areas of Interest
3 more
Logic and Philosophy of Logic |
Mathematical Logic |
The Infinite |
Set Theory |
Modal Logic |
Game Theory |
Theory of Computation |
Hypercomputation |