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Set Theory and Structures

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2019

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Friedman, Sy-David

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https://doi.org/10.1007/978-3-030-15655-8_10
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CENTRONE, Stefania, ed., Deborah KANT, ed., Deniz SARIKAYA, ed.. Reflections on the Foundations of Mathematics : Univalent Foundations, Set Theory and General Thoughts. Cham: Springer, 2019, pp. 223-253. Synthese Library. 407. ISBN 978-3-030-15654-1. Available under: doi: 10.1007/978-3-030-15655-8_10

Zusammenfassung

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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100 Philosophie

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ISO 690BARTON, Neil, Sy-David FRIEDMAN, 2019. Set Theory and Structures. In: CENTRONE, Stefania, ed., Deborah KANT, ed., Deniz SARIKAYA, ed.. Reflections on the Foundations of Mathematics : Univalent Foundations, Set Theory and General Thoughts. Cham: Springer, 2019, pp. 223-253. Synthese Library. 407. ISBN 978-3-030-15654-1. Available under: doi: 10.1007/978-3-030-15655-8_10
BibTex
@incollection{Barton2019Theor-52678,
  year={2019},
  doi={10.1007/978-3-030-15655-8_10},
  title={Set Theory and Structures},
  number={407},
  isbn={978-3-030-15654-1},
  publisher={Springer},
  address={Cham},
  series={Synthese Library},
  booktitle={Reflections on the Foundations of Mathematics : Univalent Foundations, Set Theory and General Thoughts},
  pages={223--253},
  editor={Centrone, Stefania and Kant, Deborah and Sarikaya, Deniz},
  author={Barton, Neil and Friedman, Sy-David}
}
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