It is long known that Lawvere's theory in The category of categories as foundations of mathematics A does not work, as indicated in Ishell's review . 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 ) and suggests adding CDT to the axioms.
Our starting point was the claim in  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  are all nontheorems. The remaining results indicated in  concerning BT are provable. However, as the Predicative Functor Construction Schema—PFCS—are justified in  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.
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