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Set Theory and Structures

  • Neil BartonEmail author
  • Sy-David Friedman
Chapter
Part of the Synthese Library book series (SYLI, volume 407)

Abstract

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a ‘structural’ perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Kurt Gödel Research Center for Mathematical Logic (KGRC)ViennaAustria

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