59 found
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  1. The modal logic of set-theoretic potentialism and the potentialist maximality principles.Joel David Hamkins & Øystein Linnebo - 2022 - Review of Symbolic Logic 15 (1):1-35.
    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-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...)
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  2. 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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  3.  57
    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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  4.  66
    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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  5. 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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  6.  29
    Lectures on the philosophy of mathematics.Joel David Hamkins - 2020 - Cambridge, Massachusetts: The MIT Press.
    An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and (...)
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  7. 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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  8.  38
    The exact strength of the class forcing theorem.Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlicht & Kameryn J. Williams - 2020 - Journal of Symbolic Logic 85 (3):869-905.
    The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...)
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  9.  43
    What is the theory without power set?Victoria Gitman, Joel David Hamkins & Thomas A. Johnstone - 2016 - Mathematical Logic Quarterly 62 (4-5):391-406.
    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 are sets of (...)
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  10. (1 other version)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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  11.  37
    Resurrection axioms and uplifting cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.
    We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.
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  12.  28
    The σ1-definable universal finite sequence.Joel David Hamkins & Kameryn J. Williams - 2022 - Journal of Symbolic Logic 87 (2):783-801.
    We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...)
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  13.  25
    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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  14. 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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  15. 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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  16.  73
    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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  17.  72
    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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  18.  46
    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.
    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 superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...)
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  19.  97
    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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  20.  84
    Every countable model of set theory embeds into its own constructible universe.Joel David Hamkins - 2013 - Journal of Mathematical Logic 13 (2):1350006.
    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 (...)
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  21.  98
    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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  22.  61
    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.
  23.  34
    Bi-interpretation in weak set theories.Alfredo Roque Freire & Joel David Hamkins - 2021 - Journal of Symbolic Logic 86 (2):609-634.
    In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo–Fraenkel set theory $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\mathrm {ZFC}^{-}$ that are bi-interpretable, but not isomorphic—even $\langle H_{\omega _1},\in (...)
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  24.  29
    Reflection in Second-Order Set Theory with Abundant Urelements Bi-Interprets a Supercompact Cardinal.Joel David Hamkins & Bokai Yao - forthcoming - Journal of Symbolic Logic:1-36.
    After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal κ is supercompact if and only if every Π11 sentence true in a structure M (of any size) containing κ in (...)
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  25.  38
    Set-theoretic blockchains.Miha E. Habič, Joel David Hamkins, Lukas Daniel Klausner, Jonathan Verner & Kameryn J. Williams - 2019 - Archive for Mathematical Logic 58 (7-8):965-997.
    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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  26.  86
    Diamond (on the regulars) can fail at any strongly unfoldable cardinal.Mirna Džamonja & Joel David Hamkins - 2006 - Annals of Pure and Applied Logic 144 (1-3):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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  27.  97
    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 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. If there is a (...)
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  28.  40
    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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  29.  54
    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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  30. 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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  31.  24
    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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  32. Inner-Model Reflection Principles.Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz & Ralf Schindler - 2020 - Studia Logica 108 (3):573-595.
    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 (...)
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  33.  73
    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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  34. With infinite utility, more needn't be better.Joel David Hamkins & Barbara Montero - 2000 - Australasian Journal of Philosophy 78 (2):231 – 240.
  35.  70
    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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  36.  52
    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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  37.  38
    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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  38.  43
    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.
    We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}-supercompact, for any desired θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or (...)
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  39.  20
    Modal Model Theory.Joel David Hamkins & Wojciech Aleksander Wołoszyn - 2024 - Notre Dame Journal of Formal Logic 65 (1):1-37.
    We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) (...)
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  40.  60
    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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  41. 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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  42.  80
    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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  43.  23
    When does every definable nonempty set have a definable element?François G. Dorais & Joel David Hamkins - 2019 - Mathematical Logic Quarterly 65 (4):407-411.
    The assertion that every definable set has a definable element is equivalent over to the principle, and indeed, we prove, so is the assertion merely that every Π2‐definable set has an ordinal‐definable element. Meanwhile, every model of has a forcing extension satisfying in which every Σ2‐definable set has an ordinal‐definable element. Similar results hold for and and other natural instances of.
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  44.  48
    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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  45.  57
    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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  46.  15
    Zfc proves that the class of ordinals is not weakly compact for definable classes.Ali Enayat & Joel David Hamkins - 2018 - Journal of Symbolic Logic 83 (1):146-164.
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  47.  12
    Every Countable Model of Arithmetic or Set Theory has a Pointwise-Definable End Extension.Joel David Hamkins - forthcoming - Kriterion – Journal of Philosophy.
    According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of (...)
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  48.  15
    Infinite Wordle and the mastermind numbers.Joel David Hamkins - forthcoming - Mathematical Logic Quarterly.
    I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game‐theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of n letters, including infinite words (...)
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  49.  45
    Choiceless large cardinals and set‐theoretic potentialism.Raffaella Cutolo & Joel David Hamkins - 2022 - Mathematical Logic Quarterly 68 (4):409-415.
    We define a potentialist system of ‐structures, i.e., a collection of possible worlds in the language of connected by a binary accessibility relation, achieving a potentialist account of the full background set‐theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the (...)
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  50.  69
    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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