Non Σ_n axiomatizable almost strongly minimal theories

Extract

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ₀-categorical almost strongly minimal theory is Σ_n axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ₀-categorical and almost strongly minimal, then T is Σ_n axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ₀-categorical almost strongly minimal theory which is not Σ_n axiomatizable.

Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.

We will prove the following result.

Theorem. For every n there is an almost strongly minimal ℵ₀-categorical theory T with models M and N such that N is Σ_n elementary but not Σ_{n + 1} elementary.

To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].

Theorem. If T is a Σ_n axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σ_n elementary extension of M, then N is an elementary extension of M.