Notes on Quasiminimality and Excellence

Abstract

Zilber's proposes [60] to prove 'canonicity results for pseudo-analytic' structures. Informally, 'canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power' while 'pseudoanalytic means the model of power 2 ℵ0 can be taken as a reduct of an expansion of the complex numbers by analytic functions'. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [31], this line was taken up in a long series of works by Shelah. We place Zilber's work in this context. The second direction stems from Hrushovski's construction of a counterexample to Zilber's conjecture that every strongly minimal set is 'trivial', 'vector space-like', or 'field-like'. This construction turns out to be very concrete example of the Abstract Elementary Classes which arose in Shelah's analysis. This paper examines the intertwining of these three themes.The study of (C, +, ·, exp) leads one immediately to some extension of first order logic; the integers with all their arithmetic are first order definable in (C, +, ·, exp). Thus, the first order theory of complex exponentiation is horribly complicated; it is certainly unstable and so can't be first order categorical. One solution is to use infinitary logic to pin down the pathology. Insist that the kernel of the exponential map is fixed as a single copy of the integers while allowing the rest of the structure to grow. We describe in Section 5 Zilber's program to show, modulo certain (very serious) algebraic hypotheses, that (C, +, ·, exp) can be axiomatized by a categorical L ω 1 ,ω -sentence.Of course, the extension from first order logic causes the failure of the compactness theorem. (E.g., it is easy to write a sentence in L ω 1 ,ω whose only model is the natural numbers with successor). But there are some more subtle losses. In first order logic, a type can be given as a syntactic object -a consistent set of formulas. Consider the theory T of a dense linear order without endpoints, a unary predicate P (x) which is dense and codense, and an infinite set of constants arranged in order type ω + ω * . Let K be class of all models of T which omit the type of a pair of points, which are both in the cut determined by the constants. Now consider the types p and q which are satisfied by a point in the cut, which is in P or in ¬P respectively. Now p and q are each satisfiable in a member of K but they are not simultaneously satisfiable. This failure of amalgamation shows that a more subtle notion than consistency is needed to describe types.We took 'canonical' above as meaning 'categorical in uncountable cardinalities'. The analysis of first order theories categorical in power is based on first studying strongly minimal sets: every definable subset is finite or cofinite. A natural generalization of this, particularly since it holds of simply defined subsets of (C, +, ·, exp), is to consider structur...

v*-ALGEBRAS, INDEPENDENCE ALGEBRAS AND LOGIC

“…For the latter, see also [14] where it is shown that the excellence condition yields directly something like an M-independence (where, however, the morphisms are not just homomorphisms).…”

The Vaught Conjecture: Do Uncountable Models Count?

“…This 'reduction' is not direct. In order to deduce categoricity for an arbitrary L ω 1 ,ω -sentence, stronger results than transfer of categoricity must be proved for complete L ω 1 ,ω sentences ( [18] expounded in [3,1]).…”

“…This leads us to some natural generalization of Theorem 3.3. The notion of an excellent class [18,19,3,23] plays a crucial role in the model theory of infinitary logic. These questions pose two difficulties.…”

Abstract elementary classes: some answers, more questions

“…Excellence requires a notion of independence; essentially excellence consists in requiring the existence of 'prime models' over independent n-cubes. See [Bal04] for an intuitive introduction. Grossberg and Hart [GH89] prove a 'main gap' theorem in their context.…”

“…It is substantially easier if ψ is assumed to have arbitrarily large models ([Bal] VII.2) than without that hypothesis ( [Bal] VII.3). The difficult case is carried out in full in [She75,She83a]; the easier case is hinted at in [She75,She83a] but spelled out in the expository [Bal04,Bal]. In either case a notion of stability (counting the number of types) is used to obtain even the completeness result.…”