Lawvere's basic theory of the category of categories

Abstract

It is long known that Lawvere's theory in The category of categories as foundations of mathematics A[1] does not work, as indicated in Ishell's review [0]. Isbell there gives a counterexample that CDT—Category Description Theorem—[1, p. 15] is in fact not a theorem of BT (the Basic Theory of [1]) and suggests adding CDT to the axioms.Our starting point was the claim in [1] that “the basic theory needs no explicit axiom of infinity.” We define a model ℳ of BT in which all categories are finite. In particular, the “monoid of nonnegative integers N” coincides in ℳ with the terminal object 1. We study ℳ in some detail in order to establish the true status of various “theorems” or “metatheorems” of BT: The metatheorem of [1, p. 11] saying that the discrete categories form a category of sets, CDT, the theorem on p. 15, and the theorem on p. 16 of [1] are all nontheorems. The remaining results indicated in [1] concerning BT are provable. However, as the Predicative Functor Construction Schema—PFCS—are justified in [1] by using the “metatheorem” and CDT, we provide a proof of these two schemata by showing that the discrete categories of BT (or of convenient extensions of BT) form a two-valued Boolean topos.