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360Infinite time Turing machinesMinds and Machines 12 (4): 567-604. 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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316Inner-Model Reflection PrinciplesStudia Logica 108 (3): 573-595. 2020.We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width re…Read more
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299The 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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226Gap 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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164The modal logic of set-theoretic potentialism and the potentialist maximality principlesReview of Symbolic Logic 15 (1): 1-35. 2022.We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-…Read more
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163Is 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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147Utilitarianism in Infinite WorldsUtilitas 12 (1): 91. 2000.Recently in the philosophical literature there has been some effort made to understand the proper application of the theory of utilitarianism to worlds in which there are infinitely many bearers of utility. Here, we point out that one of the best, most inclusive principles proposed to date contradicts fundamental utilitarian ideas, such as the idea that adding more utility makes a better world
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141A Natural Model of the Multiverse AxiomsNotre Dame Journal of Formal Logic 51 (4): 475-484. 2010.If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins
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130Infinite time Turing machinesJournal of Symbolic Logic 65 (2): 567-604. 2000.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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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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100The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinalMathematical Logic Quarterly 51 (5): 493-498. 2005.The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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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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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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95Indestructibility and the level-by-level agreement between strong compactness and supercompactnessJournal of Symbolic Logic 67 (2): 820-840. 2002.Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can
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92Inner models with large cardinal features usually obtained by forcingArchive for Mathematical Logic 51 (3-4): 257-283. 2012.We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compac…Read more
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86Indestructible 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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83Canonical seeds and Prikry treesJournal of Symbolic Logic 62 (2): 373-396. 1997.Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's
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80Pointwise 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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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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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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73Unfoldable 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 κ can be made indestructible by the forcing to add any number of Cohen subsets to κ
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69Every countable model of set theory embeds into its own constructible universeJournal of Mathematical Logic 13 (2): 1350006. 2013.The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordere…Read more
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68Degrees of rigidity for Souslin treesJournal of Symbolic Logic 74 (2): 423-454. 2009.We investigate various strong notions of rigidity for Souslin trees, separating them under ♢ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ♢ that there is a group whose automorphism tower is highly malleable by forcing
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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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61Destruction or preservation as you like itAnnals of Pure and Applied Logic 91 (2-3): 191-229. 1998.The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of …Read more
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61Generalizations of the Kunen inconsistencyAnnals of Pure and Applied Logic 163 (12): 1872-1890. 2012.We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one set-forcing ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed f…Read more
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59The rigid relation principle, a new weak choice principleMathematical Logic Quarterly 58 (6): 394-398. 2012.The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals is provable without the axiom of choice
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58P^f NP^f for almost all fMathematical Logic Quarterly 49 (5): 536. 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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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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57Small 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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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 |