Sets, classes, and categories

British Journal for the Philosophy of Science 52 (3):539-573 (2001)
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Abstract

This paper, accessible for a general philosophical audience having only some fleeting acquaintance with set-theory and category-theory, concerns the philosophy of mathematics, specifically the bearing of category-theory on the foundations of mathematics. We argue for six claims. (I) A founding theory for category-theory based on the primitive concept of a set or a class is worthwile to pursue. (II) The extant set-theoretical founding theories for category-theory are conceptually flawed. (III) The conceptual distinction between a set and a class can be seen to be formally codified in Ackermann's axiomatisation of set-theory. (IV) A slight but significant deductive extension of Ackermann's theory of sets and classes founds Cantorian set-theory as well as category-theory, and therefore can pass as a founding theory of the whole of mathematics. (V) The extended theory does not suffer from the conceptual flaws of the extant set-theoretical founding theories. (VI) The extended theory is not only conceptually but also logically superior to the competing set-theories because its consistency can be proved on the basis of weaker assumptions than the consistency of the competition.

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F. A. Muller
Erasmus University Rotterdam

Citations of this work

Maddy On The Multiverse.Claudio Ternullo - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Berlin: Springer Verlag. pp. 43-78.
When Do Some Things Form a Set?Simon Hewitt - 2015 - Philosophia Mathematica 23 (3):311-337.
Classes, why and how.Thomas Schindler - 2019 - Philosophical Studies 176 (2):407-435.
Deflating skolem.F. A. Muller - 2005 - Synthese 143 (3):223-253.

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References found in this work

Introduction to Mathematical Philosophy.Bertrand Russell - 1919 - Revue Philosophique de la France Et de l'Etranger 89:465-466.
Collected Papers on Mathematics, Logic, and Philosophy. [REVIEW]P. Cortois - 1988 - Tijdschrift Voor Filosofie 50 (3):558-559.
From absolute to local mathematics.J. L. Bell - 1986 - Synthese 69 (3):409 - 426.

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