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  1. Intuitionistic sets and numbers: small set theory and Heyting arithmetic.Stewart Shapiro, Charles McCarty & Michael Rathjen - forthcoming - Archive for Mathematical Logic.
    It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. (...)
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  • Bounded finite set theory.Laurence Kirby - 2021 - Mathematical Logic Quarterly 67 (2):149-163.
    We define an axiom schema for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic,, and use some of its basic model theory to establish some independence results for various axioms of set theory over. Then we ask: given a model M of, is there a model of whose ordinal arithmetic is isomorphic to M? We show that the answer is yes if.
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  • A hierarchy of hereditarily finite sets.Laurence Kirby - 2008 - Archive for Mathematical Logic 47 (2):143-157.
    This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy.
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  • On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the (...)
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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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  • Constructive Ackermann's interpretation.Hanul Jeon - 2022 - Annals of Pure and Applied Logic 173 (5):103086.
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  • Substandard models of finite set theory.Laurence Kirby - 2010 - Mathematical Logic Quarterly 56 (6):631-642.
    A survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set-theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω.
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  • Addition and multiplication of sets.Laurence Kirby - 2007 - Mathematical Logic Quarterly 53 (1):52-65.
    Ordinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non-empty elements of any model of set theory can be partitioned into infinitely many submodels, each isomorphic to (...)
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  • Digraph parameters and finite set arithmetic.Laurence Kirby - 2015 - Mathematical Logic Quarterly 61 (4-5):250-262.
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  • Finitary Set Theory.Laurence Kirby - 2009 - Notre Dame Journal of Formal Logic 50 (3):227-244.
    I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.
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  • Finiteness Axioms on Fragments of Intuitionistic Set Theory.Riccardo Camerlo - 2007 - Notre Dame Journal of Formal Logic 48 (4):473-488.
    It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of hereditarily finite sets, developed by Previale in "Induction and foundation in the theory of hereditarily finite sets." This solves some problems stated there. The (...)
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