A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every N0-stable, N0-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an

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... licit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 132 EHUD HRUSHOVSKI The theorem of § 1 implies that there are only countably many KQ-categorical, K0-stable theories, but gives no explicit structure theorem. In §4 we attempt to give one, assuming that the coordinatizing strongly minimal set is disintegrated. (Most of the results require only the weaker assumption that no affine spaces are present.) We give an explicit construction of a certain class of disintegrated totally categorical structures. It falls short of containing all of them, but not by much: every disintegrated totally categorical structure expands to a member of the class by simply naming a finite set of parameters. (In particular, we obtain a direct proof of the original Ahlbrandt-Ziegler result in this case.) We also show that the information lost in the naming of parameters is controlled by nilpotent automorphism groups. As a corollary, the automorphism group of such a structure has a finite Jordan-Holder decomposition (as a topological group) in which each of the components are of a known type. Aside from the basic results that set the groundwork (Morley, Baldwin-Lachlan, Zilber, Cherlin-Harrington-Lachlan), this paper has several direct precursors. The basic idea of §2 originated with [A-Z], where the conjecture was proved in the almost strongly minimal case. Lachlan then proved the finite language result for structures of disintegrated type. The common generalization was then achieved by Cherlin, who proved both finite language and quasi-finite axiomatizability for the class of N0-categorical, N0-stable structures of modular type. (One needs to redefine quasi-finite axiomatizability in this context to take into account the existence of several dimensions.) The basic setup for the general proof was already present there. §3 generalizes a result from [L]. §4 can also be viewed as strengthening [L], for example the result there that every totally categorical structure of disintegrated type is interpretable in a dense linear ordering. It seems clear that every K0-categorical, N0-stable structure of modular type is interpretable in a finite disjoint union of universal locally lexicographically ordered vector spaces over finite fields; see [T]. This will not be pursued. The notation is from [CHL], and the basic facts proved or assumed there are assumed here also, (d x)<p(x ,y) denotes the /z-definition of <p . Greg Cherlin helped with this paper in many ways other than the obvious debt to his [C]; I would like to thank him. Finite language and axiomatizability Theorem 2.1. (a) Let M be ^-categorical, unstable. Then M admits a finite language. (b) If a finite language is chosen for M, then the theory of M is quasi-finitely axiomatizable. A theory T is quasi-finitely axiomatizable iff there exist a finite TQ c T and a finite number of axiom schemes of infinity true in models of T, which together axiomatize T. An axiom scheme of infinity is a collection of axioms that assert for certain formulas ç and D that in any model of T, if |= <p(b) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use