Set theory and structures. (English) Zbl 1528.03011
Centrone, Stefania (ed.) et al., Reflections on the foundations of mathematics. Univalent foundations, set theory and general thoughts. Based on the conference on foundations of mathematics: univalent foundations and set theory, FOMUS, Bielefeld, Germany, July 18–23, 2016. Cham: Springer. Synth. Libr. 407, 223-253 (2019).
Summary: 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.
For the entire collection see [Zbl 1502.03003].
For the entire collection see [Zbl 1502.03003].
MSC:
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 00A30 | Philosophy of mathematics |
| 03E30 | Axiomatics of classical set theory and its fragments |
| 18A15 | Foundations, relations to logic and deductive systems |
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