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  1. Remarks on Ontological Dependence in Set Theory.Thomas Macaulay Ferguson - 2016 - Australasian Journal of Logic 13 (3):41-57.
    In a recent paper, John Wigglesworth explicates the notion of a set's being grounded in or ontologically depending on its members by the modal statement that in any world, that a set exists in that world entails that its members exist as well. After suggesting that variable-domain S5 captures an appropriate account of metaphysical necessity, Wigglesworth purports to prove that in any set theory satisfying the axiom Extensionality this condition holds, that is, that sets ontologically depend on their members with (...)
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  • Typicality À la Russell in Set Theory.Athanassios Tzouvaras - 2022 - Notre Dame Journal of Formal Logic 63 (2).
  • 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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  • 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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  • When Does Every Definable Nonempty Set Have a Definable Element?François G. Dorais & Joel David Hamkins - 2019 - Mathematical Logic Quarterly 65 (4):407-411.
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  • A Classical Way Forward for the Regularity and Normalization Problems.Alexander R. Pruss - 2021 - Synthese 199 (5-6):11769-11792.
    Bayesian epistemology has struggled with the problem of regularity: how to deal with events that in classical probability have zero probability. While the cases most discussed in the literature, such as infinite sequences of coin tosses or continuous spinners, do not actually come up in scientific practice, there are cases that do come up in science. I shall argue that these cases can be resolved without leaving the realm of classical probability, by choosing a probability measure that preserves “enough” regularity. (...)
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