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The British Journal for the Philosophy of Science 2001 52(3):539-573; doi:10.1093/bjps/52.3.539
© 2001 by British Society for the Philosophy of Science
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Sets, Classes, and Categories

F.A. Muller1

1 Institute of the History and Foundations of Mathematics and Natural Science, Utrecht University, PO Box 80.000, 3508 TA Utrecht, The Netherlands. f.a.muller@phys.uu.nl

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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