Joel David Hamkins City University of New York
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  • Faculty, City University of New York

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About me
My main research interest lies in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. I have worked particularly with forcing and large cardinals, those strong axioms of infinity, and have been particularly interested in the interaction of these two central set-theoretic concepts. I have worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess. Recently, I am preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as my work on the modal logic of forcing and set-theoretic geology.
My works
57 items found.
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  1.  6
    Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba (2016). Superstrong and Other Large Cardinals Are Never Laver Indestructible. Archive for Mathematical Logic 55 (1-2):19-35.
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  2.  13
    Joel David Hamkins & Cole Leahy (2016). Algebraicity and Implicit Definability in Set Theory. Notre Dame Journal of Formal Logic 57 (3):431-439.
    We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only (...)
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  3.  10
    Yong Cheng, Sy-David Friedman & Joel David Hamkins (2015). Large Cardinals Need Not Be Large in HOD. Annals of Pure and Applied Logic 166 (11):1186-1198.
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  4.  13
    Brent Cody, Moti Gitik, Joel David Hamkins & Jason A. Schanker (2015). The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly $${\Theta}$$ Θ -Supercompact. Archive for Mathematical Logic 54 (5-6):491-510.
  5.  12
    Gunter Fuchs, Joel David Hamkins & Jonas Reitz (2015). Set-Theoretic Geology. Annals of Pure and Applied Logic 166 (4):464-501.
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  6.  31
    Joel David Hamkins (2015). Is the Dream Solution of the Continuum Hypothesis Attainable? Notre Dame Journal of Formal Logic 56 (1):135-145.
    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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  7.  7
    Joel David Hamkins & Thomas A. Johnstone (2014). Resurrection Axioms and Uplifting Cardinals. Archive for Mathematical Logic 53 (3-4):463-485.
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  8.  9
    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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  9.  14
    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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  10.  32
    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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  11.  53
    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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  12.  27
    Joel David Hamkins, Greg Kirmayer & Norman Lewis Perlmutter (2012). Generalizations of the Kunen Inconsistency. Annals of Pure and Applied Logic 163 (12):1872-1890.
    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 (...)
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  13.  30
    Joel David Hamkins & Justin Palumbo (2012). The Rigid Relation Principle, a New Weak Choice Principle. Mathematical Logic Quarterly 58 (6):394-398.
    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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  14.  19
    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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  15.  47
    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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  16.  41
    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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  17.  13
    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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  18.  11
    Joel D. Hamkins (2009). Tall Cardinals. Mathematical Logic Quarterly 55 (1):68-86.
    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 (...)
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  19.  21
    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.
    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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  20.  10
    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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  21.  22
    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.
    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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  22. J. Hamkins & A. Myasnikov (2006). The Halting Problem is Almost Always Decidable. Notre Dame Journal of Formal Logic 47 (4):515-524.
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  23.  20
    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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  24. Joel Hamkins (2005). Lectures in Logic and Set Theory, Vols. I & II. [REVIEW] Bulletin of Symbolic Logic 11 (2):241-242.
     
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  25.  8
    Joel D. Hamkins & W. Hugh Woodin (2005). The Necessary Maximality Principle for C. C. C. Forcing is Equiconsistent with a Weakly Compact Cardinal. Mathematical Logic Quarterly 51 (5):493-498.
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal.
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  26.  27
    Joel D. Hamkins & W. Hugh Woodin (2005). The Necessary Maximality Principle for Ccc Forcing is Equiconsistent with a Weakly Compact Cardinal. Mathematical Logic Quarterly 51 (5):493-498.
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal.
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  27.  1
    Joel David Hamkins (2005). Lectures in Logic and Set Theory, Volumes 1 and 2. Bulletin of Symbolic Logic 11 (2):241-243.
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  28.  10
    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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  29. 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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  30.  26
    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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  31. J. D. Hamkins (2003). P^F NP^F for Almost All F. Mathematical Logic Quarterly 49 (5):536.
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  32. J. D. Hamkins & P. D. Welch (2003). Pf= NPf Almost Everywhere. Mathematical Logic Quarterly 49 (5):536-540.
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  33.  28
    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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  34.  15
    Philip D. Welch & Joel David Hamkins (2003). Pf ≠ NPf for Almost All F. Mathematical Logic Quarterly 49 (5):536.
    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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  35.  25
    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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  36.  39
    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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  37.  43
    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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  38.  14
    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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  39.  7
    J. D. Hamkins & A. W. Apter (2001). Indestructible Weakly Compact Cardinals and the Necessity of Supercompactness for Certain Proof Schemata. Mathematical Logic Quarterly 47 (4):563-572.
    We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.
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  40.  17
    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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  41.  26
    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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  42.  12
    D. E. Seabold & J. D. Hamkins (2001). Infinite Time Turing Machines With Only One Tape. Mathematical Logic Quarterly 47 (2):271-287.
    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 (...)
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  43.  21
    Joel David Hamkins (2000). The Lottery Preparation. Annals of Pure and Applied Logic 101 (2-3):103-146.
    The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...)
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  44.  45
    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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  45.  27
    Joel David Hamkins & Barbara Montero (2000). With Infinite Utility, More Needn't Be Better. Australasian Journal of Philosophy 78 (2):231 – 240.
  46.  40
    Joel David Hamkins & Barbara Montero (2000). Utilitarianism in Infinite Worlds. Utilitas 12 (1):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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  47.  22
    Joel David Hamkins & Simon Thomas (2000). Changing the Heights of Automorphism Towers. Annals of Pure and Applied Logic 102 (1-2):139-157.
    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<λ, there exists a centreless group G such that τ=α; and if β is any ordinal such that 1β<λ, then there exists a notion of forcing , which preserves cofinalities and cardinalities, such that τ=β in the corresponding generic extension.
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  48. 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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  49. C. Butz, P. Johnstone, J. Gallier, J. D. Hamkins, B. Khoussaiuov, H. Lombardi & C. Raffalli (1998). Andrkka, H., Givant, S., Mikulb, S., Ntmeti, I. And Simon, A. Annals of Pure and Applied Logic 91:271.
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  50.  23
    Joel David Hamkins (1998). Destruction or Preservation as You Like It. Annals of Pure and Applied Logic 91 (2-3):191-229.
    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 what I call (...)
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  51.  19
    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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  52.  25
    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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  53. Joel David Hamkins & Saharon Shelah (1998). Superdestructibility: A Dual to the Laver Preparation. Journal of Symbolic Logic 63:549-554.
     
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  54.  24
    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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  55.  1
    Joel David Hamkins (1997). Moschovakis Yiannis N.. Notes on Set Theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York, Berlin, Heidelberg, Etc., 1994, Xiv + 272 Pp. [REVIEW] Journal of Symbolic Logic 62 (4):1493-1494.
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  56.  8
    Joel David Hamkins (1997). Review: Yiannis N. Moschovakis, Notes on Set Theory. [REVIEW] Journal of Symbolic Logic 62 (4):1493-1494.
  57.  14
    Joel Hamkins (1994). Fragile Measurability. Journal of Symbolic Logic 59 (1):262-282.
    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 $\kappa^{<\kappa}$ and κ+, but not P(κ), destroys the measurability of κ, even if κ is initially supercompact, strong, or if I1(κ) holds. Obtained as an application of some general lifting theorems, this result is an (...)
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