Sets, classes, and categories

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. 1 Exordium 1.1 Motivation 1.2 Overview 1.3 Terminology 2 Category-theory versus set-theory 2.1 The downfall of set-theory 2.2 The resurrection of set-theory 2.3 Criticism of the resurrection 3 Sets and classes 3.1 Sets as classes 3.2 Requirements for a founding theory 4 A theory of sets and classes 4.1 The language of sets and classes 4.2 Axioms for sets and classes 4.3 Logical respect 4.4 The Levy-Vaught Theorem 5 Conclusions and reflections 5.1 Neo-Bourbakistan 5.2 Classes and membership: superfluous or unavoidable?