Chains of of Theories
A talk in the Istanbul Model Theory Seminar, February 28, 2013.
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.
8 pp., A5 paper: