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33Lectures in Logic and Set Theory, volumes 1 and 2 (review)Bulletin of Symbolic Logic 11 (2): 241-243. 2005.
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87Indestructible Strong UnfoldabilityNotre Dame Journal of Formal Logic 51 (3): 291-321. 2010.Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all
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6Canonical Seeds and Prikry TreesJournal of Symbolic Logic 62 (2): 373-396. 1997.Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjectur…Read more
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4Unfoldable Cardinals and the GCHJournal of Symbolic Logic 66 (3): 1186-1198. 2001.Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal $\kappa$ can be made indestructible by the forcing to add any number of Cohen subsets to $\kappa$.
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39Tall cardinalsMathematical Logic Quarterly 55 (1): 68-86. 2009.A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j > θ and Mκ ⊆ M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal κ can be made indestructible by a variety o…Read more
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42Superstrong and other large cardinals are never Laver indestructibleArchive for Mathematical Logic 55 (1-2): 19-35. 2016.Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdest…Read more
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48Post's problem for supertasks has both positive and negative solutionsArchive for Mathematical Logic 41 (6): 507-523. 2002.The infinite time Turing machine analogue of Post's problem, the question whether there are semi-decidable supertask degrees between 0 and the supertask jump 0∇, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0∇, but in the context of sets of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ a kind of transfinite-injury construction which generalizes canonically to …Read more
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26Infinite Time Turing MachinesMinds and Machines 12 (4): 521-539. 2002.Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.
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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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38Infinite Time Turing Machines With Only One TapeMathematical Logic Quarterly 47 (2): 271-287. 2001.Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for partial functions f : ℝ → ℕ, the same class of computable functions. Nevertheless, there are infinite time computable functions f : ℝ → ℝ that are not one-tape computable, and so the two models of infinitary computation are not equivalent. Surprisingl…Read more
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96A 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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303The 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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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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63Superdestructibility: 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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81Pointwise 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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164Is 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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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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101With infinite utility, more needn't be betterAustralasian Journal of Philosophy 78 (2). 2000.This Article does not have an abstract
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36What 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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52Large cardinals need not be large in HODAnnals of Pure and Applied Logic 166 (11): 1186-1198. 2015.
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35Post’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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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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57The 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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73New 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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96Exactly 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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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 |