This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirty-five years. The first three were “Set-theoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have

By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (Zermelo-Frankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “Zermelo-Fraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The Frege-Russell definition of the natural numbers....... 4

Abstract. In the informal unlimited theory of structures and (particularly) categories, one considers unrestricted statements concerning structures such as that the substructure relation on all structures of a given kind forms a partially ordered structure. or that the collection of all categories forms a category with arbitrary These sorts of propositio~s are not accounted for difunctors as its morphisms. The aim of the present work is to give a founrectly by currently accepted means. dation for the theory of structures including such unlimited statements-more or less as they are presented to us-by means of certain formal systems. The theories studied here are based on an extension of Quinels idea of stratification. Their use is justified bya consistency proof. adapting methods of Jensen. These systems are successful for the basic aim to a considerable extent. but they suffer a specific defect which prevents them from being fully successful. Some possible alternatives are also suggested.

Symmetry motivates a new consistent fragment of NF and an extension of NF with