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Joel David Hamkins
Oxford University
  1. The Set-Theoretic Multiverse.Joel David Hamkins - 2012 - 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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  2.  24
    Set-Theoretic Geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
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  3.  39
    The Lottery Preparation.Joel David Hamkins - 2000 - 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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  4. Gap Forcing: Generalizing the Lévy-Solovay Theorem.Joel David Hamkins - 1999 - 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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  5.  13
    Resurrection Axioms and Uplifting Cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.
  6.  5
    What is the Theory ZFC Without Power Set?Victoria Gitman, Joel David Hamkins & Thomas A. Johnstone - 2016 - Mathematical Logic Quarterly 62 (4-5):391-406.
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  7.  9
    Inner-Model Reflection Principles.Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz & Ralf Schindler - forthcoming - Studia Logica:1-23.
    We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \\) 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 \. A stronger principle, the ground-model reflection principle, asserts that any such \\) 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 reflection in (...)
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  8.  5
    Strongly Uplifting Cardinals and the Boldface Resurrection Axioms.Joel David Hamkins & Thomas A. Johnstone - 2017 - Archive for Mathematical Logic 56 (7-8):1115-1133.
    We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
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  9.  66
    Infinite Time Turing Machines.Joel David Hamkins - 2002 - 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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  10.  49
    Indestructibility and the Level-by-Level Agreement Between Strong Compactness and Supercompactness.Arthur W. Apter & Joel David Hamkins - 2002 - 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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  11.  15
    George Tourlakis. 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]Joel David Hamkins - 2005 - Bulletin of Symbolic Logic 11 (2):241-243.
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  12.  32
    Generalizations of the Kunen Inconsistency.Joel David Hamkins, Greg Kirmayer & Norman Lewis Perlmutter - 2012 - 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.  69
    A Simple Maximality Principle.Joel David Hamkins - 2003 - 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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  14.  48
    Indestructible Strong Unfoldability.Joel David Hamkins & Thomas A. Johnstone - 2010 - 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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  15.  84
    Is the Dream Solution of the Continuum Hypothesis Attainable?Joel David Hamkins - 2015 - 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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  16.  8
    Superstrong and Other Large Cardinals Are Never Laver Indestructible.Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba - 2016 - Archive for Mathematical Logic 55 (1-2):19-35.
  17.  45
    Inner Models with Large Cardinal Features Usually Obtained by Forcing.Arthur W. Apter, Victoria Gitman & Joel David Hamkins - 2012 - 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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  18.  35
    Destruction or Preservation as You Like It.Joel David Hamkins - 1998 - 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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  19.  38
    Every Countable Model of Set Theory Embeds Into its Own Constructible Universe.Joel David Hamkins - 2013 - Journal of Mathematical Logic 13 (2):1350006.
  20.  8
    Yiannis N. Moschovakis. Notes on Set Theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York, Berlin, Heidelberg, Etc., 1994, Xiv + 272 Pp. [REVIEW]Joel David Hamkins - 1997 - Journal of Symbolic Logic 62 (4):1493-1494.
  21.  40
    Diamond (on the Regulars) Can Fail at Any Strongly Unfoldable Cardinal.Mirna Džamonja & Joel David Hamkins - 2006 - 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.  22
    The Wholeness Axioms and V=HOD.Joel David Hamkins - 2001 - 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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  23.  32
    Exactly Controlling the Non-Supercompact Strongly Compact Cardinals.Arthur W. Apter & Joel David Hamkins - 2003 - 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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  24.  69
    A Natural Model of the Multiverse Axioms.Victoria Gitman & Joel David Hamkins - 2010 - 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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  25.  60
    Infinite Time Turing Machines.Joel David Hamkins & Andy Lewis - 2000 - 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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  26.  3
    Set-Theoretic Blockchains.Miha E. Habič, Joel David Hamkins, Lukas Daniel Klausner, Jonathan Verner & Kameryn J. Williams - forthcoming - Archive for Mathematical Logic:1-33.
    Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings (...)
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  27.  67
    Canonical Seeds and Prikry Trees.Joel David Hamkins - 1997 - 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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  28.  19
    Large Cardinals Need Not Be Large in HOD.Yong Cheng, Sy-David Friedman & Joel David Hamkins - 2015 - Annals of Pure and Applied Logic 166 (11):1186-1198.
  29.  18
    The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly $${\Theta}$$ Θ -Supercompact.Brent Cody, Moti Gitik, Joel David Hamkins & Jason A. Schanker - 2015 - Archive for Mathematical Logic 54 (5-6):491-510.
  30.  38
    Small Forcing Makes Any Cardinal Superdestructible.Joel David Hamkins - 1998 - 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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  31.  28
    Superdestructibility: A Dual to Laver's Indestructibility.Joel David Hamkins & Saharon Shelah - 1998 - 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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  32.  35
    Degrees of Rigidity for Souslin Trees.Gunter Fuchs & Joel David Hamkins - 2009 - 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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  33.  28
    Changing the Heights of Automorphism Towers.Joel David Hamkins & Simon Thomas - 2000 - 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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  34. Utilitarianism in Infinite Worlds.Joel David Hamkins & Barbara Montero - 2000 - 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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  35.  25
    Post's Problem for Supertasks has Both Positive and Negative Solutions.Joel David Hamkins & Andrew Lewis - 2002 - 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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  36.  30
    Infinite Time Decidable Equivalence Relation Theory.Samuel Coskey & Joel David Hamkins - 2011 - 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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  37.  48
    With Infinite Utility, More Needn't Be Better.Joel David Hamkins & Barbara Montero - 2000 - Australasian Journal of Philosophy 78 (2):231 – 240.
  38.  35
    Unfoldable Cardinals and the GCH.Joel David Hamkins - 2001 - 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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  39.  22
    Post’s Problem for Ordinal Register Machines: An Explicit Approach.Joel David Hamkins & Russell G. Miller - 2009 - 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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  40.  37
    The Halting Problem Is Decidable on a Set of Asymptotic Probability One.Joel David Hamkins & Alexei Miasnikov - 2006 - 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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  41.  50
    New Inconsistencies in Infinite Utilitarianism: Is Every World Good, Bad or Neutral?Donniell Fishkind, Joel David Hamkins & Barbara Montero - 2002 - 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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  42.  6
    Ehrenfeucht’s Lemma in Set Theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2018 - Notre Dame Journal of Formal Logic 59 (3):355-370.
    Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. (...)
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  43.  6
    A Model of the Generic Vopěnka Principle in Which the Ordinals Are Not Mahlo.Victoria Gitman & Joel David Hamkins - 2019 - Archive for Mathematical Logic 58 (1-2):245-265.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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  44.  42
    The Rigid Relation Principle, a New Weak Choice Principle.Joel David Hamkins & Justin Palumbo - 2012 - 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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  45.  26
    Algebraicity and Implicit Definability in Set Theory.Joel David Hamkins & Cole Leahy - 2016 - 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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  46.  25
    Pointwise Definable Models of Set Theory.Joel David Hamkins, David Linetsky & Jonas Reitz - 2013 - 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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  47.  17
    Changing the Heights of Automorphism Towers by Forcing with Souslin Trees Over L.Gunter Fuchs & Joel David Hamkins - 2008 - 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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  48.  4
    Set-Theoretic Mereology.Joel David Hamkins & Makoto Kikuchi - 2016 - Logic and Logical Philosophy 25 (3):285-308.
    We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by (...)
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  49.  14
    Review: Yiannis N. Moschovakis, Notes on Set Theory. [REVIEW]Joel David Hamkins - 1997 - Journal of Symbolic Logic 62 (4):1493-1494.
  50.  3
    Incomparable Ω1 -Like Models of Set Theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2017 - Mathematical Logic Quarterly 63 (1-2):66-76.
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