Sets, Classes, and Categories
Source: British Journal for the Philosophy of Science, Volume 52, Number 3, September 2001 , pp. 539-573(35)
Publisher: Oxford University Press
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.
Document Type: Research Article
Publication date: 2001-09-01
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