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  1. Developing arithmetic in set theory without infinity: some historical remarks.Charles Parsons - 1987 - History and Philosophy of Logic 8 (2):201-213.
    In this paper some of the history of the development of arithmetic in set theory is traced, particularly with reference to the problem of avoiding the assumption of an infinite set. Although the standard method of singling out a sequence of sets to be the natural numbers goes back to Zermelo, its development was more tortuous than is generally believed. We consider the development in the light of three desiderata for a solution and argue that they can probably not all (...)
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  • (1 other version)The empty set, the Singleton, and the ordered pair.Akihiro Kanamori - 2003 - Bulletin of Symbolic Logic 9 (3):273-298.
    For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...)
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  • Superclasses in a Finite Extension of Zermelo Set Theory.Martin Kühnrich - 1978 - Mathematical Logic Quarterly 24 (31-36):539-552.
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  • Russell's Paradox and the Theory of Classes in The Principles of Mathematics.Yasushi Nomura - 2013 - Journal of the Japan Association for Philosophy of Science 41 (1):23-36.
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  • Breaking the Tie: Benacerraf’s Identification Argument Revisited.Arnon Avron & Balthasar Grabmayr - 2023 - Philosophia Mathematica 31 (1):81-103.
    Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...)
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  • A Theorem on Definitions of the Zermelo‐Neumann Ordinals.Hao Wang - 1967 - Mathematical Logic Quarterly 13 (16-18):241-250.
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  • (1 other version)Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom).Ernst Specker - 1957 - Mathematical Logic Quarterly 3 (13-20):173-210.
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  • On the Axiom of Canonicity.Jerzy Pogonowski - forthcoming - Logic and Logical Philosophy:1-29.
    The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications.
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  • Bernays and set theory.Akihiro Kanamori - 2009 - Bulletin of Symbolic Logic 15 (1):43-69.
    We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.
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  • Zermelo: Boundary numbers and domains of sets continued.Heinz-Dieter Ebbinghaus - 2006 - History and Philosophy of Logic 27 (4):285-306.
    Towards the end of his 1930 paper on boundary numbers and domains of sets Zermelo briefly discusses the questions of consistency and of the existence of an unbounded sequence of strongly inaccessible cardinals, deferring a detailed discussion to a later paper which never appeared. In a report to the Emergency Community of German Science from December 1930 about investigations in progress he mentions that some of the intended extensions of these topics had been worked out and were nearly ready for (...)
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  • A formalization of the theory of sets from the point of view of combinatory logic.Edward J. Cogan - 1955 - Mathematical Logic Quarterly 1 (3):198-240.
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