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Category theory and set theory as theories about complementary types of universals

Ellerman, David P. (2016) Category theory and set theory as theories about complementary types of universals. Logic and Logical Philosophy (Online First), 2016 (1). pp. 1-18.

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Abstract

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Ellerman, David P.david@ellerman.org
Additional Information: I checked on Romeo sherpa and this journal is RoMeo Green.
Keywords: universals, category theory, Plato's Theory of Forms, set theoretic antinomies, universal mapping properties
Subjects: General Issues > History of Philosophy of Science
Specific Sciences > Mathematics
General Issues > Realism/Anti-realism
Depositing User: David Ellerman
Date Deposited: 13 Aug 2016 19:02
Last Modified: 13 Aug 2016 19:02
Item ID: 12353
Journal or Publication Title: Logic and Logical Philosophy (Online First)
Publisher: Nicolaus Copernicus University
Official URL: http://apcz.pl/czasopisma/index.php/LLP/article/vi...
DOI or Unique Handle: https://doi.org/10.12775/LLP.2016.022
Subjects: General Issues > History of Philosophy of Science
Specific Sciences > Mathematics
General Issues > Realism/Anti-realism
Date: 5 August 2016
Page Range: pp. 1-18
Volume: 2016
Number: 1
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/12353

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