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Profile: Joel David Hamkins (City University of New York)
  1. Joel David Hamkins (2013). Every Countable Model of Set Theory Embeds Into its Own Constructible Universe. Journal of Mathematical Logic 13 (2):1350006.
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  2. Joel David Hamkins, David Linetsky & Jonas Reitz (2013). Pointwise Definable Models of Set Theory. Journal of Symbolic Logic 78 (1):139-156.
    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 pointwise definable. (...)
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  3. Arthur W. Apter, Victoria Gitman & Joel David Hamkins (2012). Inner Models with Large Cardinal Features Usually Obtained by Forcing. Archive for Mathematical Logic 51 (3-4):257-283.
    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 2 κ = κ +, another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is supercompact. (...)
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  4. Joel David Hamkins (2012). The Set-Theoretic Multiverse. Review of Symbolic Logic 5 (3):416-449.
    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 possibilities, a phenomenon that challenges the universe (...)
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  5. Joel David Hamkins, Greg Kirmayer & Norman Lewis Perlmutter (2012). Generalizations of the Kunen Inconsistency. Annals of Pure and Applied Logic 163 (12):1872-1890.
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  6. Joel David Hamkins & Justin Palumbo (2012). The Rigid Relation Principle, a New Weak Choice Principle. Mathematical Logic Quarterly 58 (6):394-398.
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  7. Samuel Coskey & Joel David Hamkins (2010). Infinite Time Decidable Equivalence Relation Theory. Notre Dame Journal of Formal Logic 52 (2):203-228.
    We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the (...)
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  8. Victoria Gitman & Joel David Hamkins (2010). A Natural Model of the Multiverse Axioms. Notre Dame Journal of Formal Logic 51 (4):475-484.
    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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  9. Joel David Hamkins & Thomas A. Johnstone (2010). Indestructible Strong Unfoldability. Notre Dame Journal of Formal Logic 51 (3):291-321.
    Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all.
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  10. Gunter Fuchs & Joel David Hamkins (2009). Degrees of Rigidity for Souslin Trees. Journal of Symbolic Logic 74 (2):423-454.
    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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  11. Joel David Hamkins & Russell G. Miller (2009). Post's Problem for Ordinal Register Machines: An Explicit Approach. Annals of Pure and Applied Logic 160 (3):302-309.
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  12. Gunter Fuchs & Joel David Hamkins (2008). Changing the Heights of Automorphism Towers by Forcing with Souslin Trees Over L. Journal of Symbolic Logic 73 (2):614 - 633.
    We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.
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  13. Mirna Džamonja & Joel David Hamkins (2006). Diamond (on the Regulars) Can Fail at Any Strongly Unfoldable Cardinal. Annals of Pure and Applied Logic 144 (1):83-95.
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  14. Joel David Hamkins & Alexei Miasnikov (2006). The Halting Problem Is Decidable on a Set of Asymptotic Probability One. Notre Dame Journal of Formal Logic 47 (4):515-524.
    The halting problem for Turing machines is decidable on a set of asymptotic probability one. The proof is sensitive to the particular computational models.
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  15. Joel David Hamkins (2005). Tourlakis George. Lectures in Logic and Set Theory, Volumes 1 and 2. Cambridge Studies in Advanced Mathematics, Vol. 83. Cambridge University Press, Cambridge, UK, 2003. Xi+ 328 and Xv+ 575 Pp. [REVIEW] Bulletin of Symbolic Logic 11 (2):241-243.
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  16. G. Tourlakis & Joel David Hamkins (2005). REVIEWS-Lectures in Logic and Set Theory, Vols. I & II. Bulletin of Symbolic Logic 11 (2):241-242.
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  17. Arthur W. Apter & Joel David Hamkins (2003). Exactly Controlling the Non-Supercompact Strongly Compact Cardinals. Journal of Symbolic Logic 68 (2):669-688.
    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 of measurable cardinals. These results improve and (...)
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  18. Joel David Hamkins (2003). A Simple Maximality Principle. Journal of Symbolic Logic 68 (2):527-550.
    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 equivalent to (...)
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  19. Philip D. Welch & Joel David Hamkins (2003). Pf ≠ NPf for Almost All F. Mathematical Logic Quarterly 49 (5):536.
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  20. Arthur W. Apter & Joel David Hamkins (2002). Indestructibility and the Level-by-Level Agreement Between Strong Compactness and Supercompactness. Journal of Symbolic Logic 67 (2):820-840.
    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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  21. Donniell Fishkind, Joel David Hamkins & Barbara Montero (2002). New Inconsistencies in Infinite Utilitarianism: Is Every World Good, Bad or Neutral? Australasian Journal of Philosophy 80 (2):178 – 190.
    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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  22. Joel David Hamkins (2002). Infinite Time Turing Machines. Minds and Machines 12 (4):567-604.
    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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  23. Joel David Hamkins & Andrew Lewis (2002). Post's Problem for Supertasks has Both Positive and Negative Solutions. Archive for Mathematical Logic 41 (6):507-523.
    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 oracles.
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  24. Joel David Hamkins (2001). The Wholeness Axioms and V=HOD. Archive for Mathematical Logic 40 (1):1-8.
    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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  25. Joel David Hamkins (2001). Unfoldable Cardinals and the GCH. Journal of Symbolic Logic 66 (3):1186-1198.
    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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  26. Joel David Hamkins (2000). The Lottery Preparation. Annals of Pure and Applied Logic 101 (2-3):103-146.
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  27. Joel David Hamkins & Andy Lewis (2000). Infinite Time Turing Machines. Journal of Symbolic Logic 65 (2):567-604.
    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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  28. Joel David Hamkins & Barbara Montero (2000). With Infinite Utility, More Needn't Be Better. Australasian Journal of Philosophy 78 (2):231 – 240.
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  29. Joel David Hamkins & Barbara Montero (2000). Utilitarianism in Infinite Worlds. Utilitas 12 (01):91-.
    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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  30. Joel David Hamkins & Simon Thomas (2000). Changing the Heights of Automorphism Towers. Annals of Pure and Applied Logic 102 (1-2):139-157.
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  31. Joel David Hamkins (1999). Gap Forcing: Generalizing the Lévy-Solovay Theorem. Bulletin of Symbolic Logic 5 (2):264-272.
    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 cardinals, and so on.
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  32. Joel David Hamkins (1998). Destruction or Preservation as You Like It. Annals of Pure and Applied Logic 91 (2-3):191-229.
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  33. Joel David Hamkins (1998). Small Forcing Makes Any Cardinal Superdestructible. Journal of Symbolic Logic 63 (1):51-58.
    Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal κ becomes superdestructible--any further <κ--closed forcing which adds a subset to κ will destroy the measurability, even the weak compactness, of κ. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.
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  34. Joel David Hamkins & Saharon Shelah (1998). Superdestructibility: A Dual to Laver's Indestructibility. Journal of Symbolic Logic 63 (2):549-554.
    After small forcing, any $ -closed forcing will destroy the supercompactness and even the strong compactness of κ.
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  35. Joel David Hamkins & Saharon Shelah (1998). Superdestructibility: A Dual to the Laver Preparation. Journal of Symbolic Logic 63:549-554.
     
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  36. Joel David Hamkins (1997). Canonical Seeds and Prikry Trees. Journal of Symbolic Logic 62 (2):373-396.
    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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  37. Joel David Hamkins (1997). Review: Yiannis N. Moschovakis, Notes on Set Theory. [REVIEW] Journal of Symbolic Logic 62 (4):1493-1494.
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