Chains of of Theories

A talk in the Istanbul Model Theory Seminar, February 28, 2013.

Abstract

I shall speak about properties of theories that are preserved under taking unions of increasing chains of theories. I talked about these things last fall, on November 29, but I shall try both:

not to assume that you were at that talk or remember it,
to say things that were not in that talk.

The main examples are the following.

For each natural number m, the theory, called m-DF, of fields with m commuting derivations has a model companion (which is the theory of the existentially closed models of m-DF, this class being elementary). Call this model-companion m-DCF. The theories m-DCF are mutually inconsistent, and therefore the union of the m-DF, which is consistent, has no model companion.

However, the theory of fields of characteristic 0 with omega-many commuting derivations does have a model-companion, which preserves the quantifier-elimination, completeness, and stability of the theories m-DCF₀, but not their omega-stability.

The theory of vector-spaces with the scalar-field as an additional sort is highly dependent on the chosen signature. If there are 2-ary, 3-ary, …, m-ary predicates for linear dependence of vectors, let the theory be m-VS. Then the existentially closed models of m-VS have dimension m, and the scalar field is algebraically closed. Thus the union of the model-companions of the m-VS is inconsistent. But the union of the m-VS has a model-companion (as I said last time), whose completions, obtained by specifying a characteristic for the field, are omega-stable. If m is at least 2, the m-ary predicate for linear dependence can be used to interpret the scalar field in the vector space (if this has dimension at least m) in such a way that, in the foregoing observations, m-VS can be replaced with the the theory of (one-sorted) abelian groups with the appropriate m-ary predicate.

Notes

8 pp., A5 paper: