Results for ' 03E70'

15 found
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  1.  13
    Cantorian Models of Predicative.Panagiotis Rouvelas - 2024 - Journal of Symbolic Logic 89 (2):637-645.
    Tangled Type Theory was introduced by Randall Holmes in [3] as a new way of approaching the consistency problem for $\mathrm {NF}$. Although the task of finding models for this theory is far from trivial (considering it is equiconsistent with $\mathrm {NF}$ ), ways of constructing models for certain fragments of it have been discovered. In this article, we present a simpler way of constructing models of predicative Tangled Type Theory and consequently of predicative $\mathrm {NF}$. In these new models (...)
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  2.  8
    Non-Tightness in Class Theory and Second-Order Arithmetic.Alfredo Roque Freire & Kameryn J. Williams - forthcoming - Journal of Symbolic Logic:1-28.
    A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including $\mathsf {PA}$ [39], $\mathsf {ZF}$, $\mathsf {Z}_2$, and $\mathsf {KM}$ [6]. In this article we extend Enayat’s investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of $\mathsf {Z}_2$ and $\mathsf {KM}$ gives non-tight theories. Specifically, we show that $\mathsf {GB}$ and $\mathsf {ACA}_0$ each admit different bi-interpretable extensions, (...)
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  3. Varieties of Class-Theoretic Potentialism.Neil Barton & Kameryn J. Williams - 2024 - Review of Symbolic Logic 17 (1):272-304.
    We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.
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  4.  70
    A Note on Choice Principles in Second-Order Logic.Benjamin Siskind, Paolo Mancosu & Stewart Shapiro - 2023 - Review of Symbolic Logic 16 (2):339-350.
    Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to (...)
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  5.  20
    An axiomatic approach to forcing in a general setting.Rodrigo A. Freire & Peter Holy - 2022 - Bulletin of Symbolic Logic 28 (3):427-450.
    The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which (...)
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  6.  7
    A Dedekind-Style Axiomatization and the Corresponding Universal Property of an Ordinal Number System.Zurab Janelidze & Ineke van der Berg - 2022 - Journal of Symbolic Logic 87 (4):1396-1418.
    In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$. The structure in our axiomatization is a triple $(O,L,s)$, where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon O\to (...)
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  7.  51
    Forcing with the Anti‐Foundation axiom.Olivier Esser - 2012 - Mathematical Logic Quarterly 58 (1-2):55-62.
    In this paper we define the forcing relation and prove its basic properties in the context of the theory ZFCA, i.e., ZFC minus the Foundation axiom and plus the Anti-Foundation axiom.
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  8.  12
    A partial model of NF with ZF.Nando Prati - 1993 - Mathematical Logic Quarterly 39 (1):274-278.
    The theory New Foundations of Quine was introduced in [14]. This theory is finitely axiomatizable as it has been proved in [9]. A similar result is shown in [8] using a system called K. Particular subsystems of NF, inspired by [8] and [9], have models in ZF. Very little is known about subsystems of NF satisfying typical properties of ZF; for example in [11] it is shown that the existence of some sets which appear naturally in ZF is an axiom (...)
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  9.  22
    A bridge between q-worlds.Benjamin Eva, Masanao Ozawa & Andreas Doering - 2021 - Review of Symbolic Logic 14 (2):447-486.
    Quantum set theory and topos quantum theory are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, (...)
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  10.  44
    Is Cantor’s Theorem a Dialetheia? Variations on a Paraconsistent Approach to Cantor’s Theorem.Uwe Petersen - forthcoming - Review of Symbolic Logic:1-18.
    The present note was prompted by Weber’s approach to proving Cantor’s theorem, i.e., the claim that the cardinality of the power set of a set is always greater than that of the set itself. While I do not contest that his proof succeeds, my point is that he neglects the possibility that by similar methods it can be shown also that no non-empty set satisfies Cantor’s theorem. In this paper unrestricted abstraction based on a cut free Gentzen type sequential calculus (...)
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  11.  15
    Paraconsistent and Paracomplete Zermelo–Fraenkel Set Theory.Yurii Khomskii & Hrafn Valtýr Oddsson - forthcoming - Review of Symbolic Logic:1-31.
    We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to (...)
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  12.  46
    Truths, Inductive Definitions, and Kripke-Platek Systems Over Set Theory.Kentaro Fujimoto - 2018 - Journal of Symbolic Logic 83 (3):868-898.
    In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.
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  13.  29
    Finality regained: A coalgebraic study of Scott-sets and multisets. [REVIEW]Giovanna D'Agostino & Albert Visser - 2002 - Archive for Mathematical Logic 41 (3):267-298.
    In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of such sets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. We will have a closer look into the connection of the iterated circular multisets and arbitrary trees. RID=""ID="" Mathematics Subject Classification (2000): 03B45, 03E65, 03E70, 18A15, (...)
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  14.  22
    Non‐well‐founded extensions of documentclass{article}usepackage{amssymb}begin{document}pagestyle{empty}$mathbf {V}$end{document}.William R. Brian - 2013 - Mathematical Logic Quarterly 59 (3):167-176.
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  15.  28
    On a positive set theory with inequality.Giacomo Lenzi - 2011 - Mathematical Logic Quarterly 57 (5):474-480.
    We introduce a quite natural Frege-style set theory, which we call Strong-Frege-2 equation image, a sort of simplification of the theory considered in 13 and 1 . We give a model of a weaker variant of equation image, called equation image, where atoms and coatoms are allowed. To construct the model we use an enumeration “almost without repetitions” of the Π11 sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of (...)
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