Abstract
Buon giorno. This is the first of eight lectures on the model theoretic notion of theory of algebraic type. Some examples of the notion are the theories of algebraically closed fields of characteristic p (p ≥ 0), real closed fields and differentially closed fields of characteristic 0. The last example is the most important for two reasons. First, it is the only one known whose complexity matches that of the general case. Second, several results about differential fields, results which hold for all theories of algebraic type, were first proved by model theoretic means.
The key definition is quite compact, but five lectures will be needed to unpack it. A theory T is said to be of algebraic type if T is complete, T is the model completion of a universal theóry, and T is quasi-totally transcendental. In the brief time left before the onset of formalities, let me indicate why the theory of algebraically closed fields of characteristic 0 (ACF0) is of algebraic type. The completeness of ACF0 means that the same first order sentences are true in all algebraically closed fields of characteristic 0. Thus a first order sentence in the language of fields is true of the complex numbers if and only if it is true of the algebraic numbers.
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Sacks, G.E. (2010). Model Theory and Applications. In: Mangani, P. (eds) Model Theory and Applications. C.I.M.E. Summer Schools, vol 69. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11121-1_1
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