Lectures on the Model Theory of Real and Complex Exponentiation
In these notes we sketch a proof of the model completeness of the real exponential field. We begin with an introduction to the various Preparation Theorems required for the proof as well as a discussion of polynomially bounded, o-minimal structures. We th
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1 Introduction We shall be interested in the structures Rexp and Cexp , the expansions of the rings of real and complex numbers respectively, by the corresponding exponential functions and, in particular, the problem of describing mathematically their definable sets. In the real case I shall give the main ideas of a proof of the theorem stating that the theory Texp of the structure Rexp is model complete. Definition 1. A theory T in a language L is called model complete if it satisfies one of the following equivalent conditions: (i) for every formula .x/ of L, there exists an existential formulas .x/ of L such that T ˆ 8x..x/ $ .x//; (ii) for all models M0 , M of T with M0 M we have that M0 41 M (i.e. existential formulas are absolute between M0 and M); (iii) for all models M0 , M of T with M0 M we have that M0 4 M (i.e. all formulas are absolute between M0 and M); (iv) for all models M of T , the LM -theory T [ Diagram.M/ is complete. (In (iv), LM is the language extending L by a constant symbol naming each element of the domain of M, and Diagram.M/ denotes the set of all quantifier-free sentences of LM that are true in M.) This is, of course, a theorem. The actual definition that gives rise to the name is (iv). I shall prove that (ii) holds for T D Texp and in this case the task can be further reduced:
A.J. Wilkie () School of Mathematics, The Alan Turing Building, University of Manchester, Manchester M13 9PL, UK e-mail: [email protected] L. van den Dries et al., Model Theory in Algebra, Analysis and Arithmetic, Lecture Notes in Mathematics 2111, DOI 10.1007/978-3-642-54936-6__3, © Springer-Verlag Berlin Heidelberg 2014
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Exercise 1. Suppose that all pairs of models M0 , M of Texp with M0 M have the property that any quasipolynomial (see below) with coefficients in M0 and having a solution (i.e. a zero) in M, also has a solution in M0 . Prove that Texp is model complete. Definition 2. Let M0 D hM0 ; :::i be a substructure of a model M D hM; :::i of Texp . Then a function (from M n to M ) that can be written in the form hx1 ; : : : ; xn i 7! P .x1 ; : : : ; xn ; exp.x1 /; : : : ; exp.xn //; where P is a polynomial (in 2n variables) with coefficients in M0 , is called a quasipolynomial with coefficients in M0 . In fact I shall only prove, in the notation of Exercise 1, that a quasipolynomial with solutions in M n has one, hb1 ; : : : ; bn i say, that is M0 -bounded, i.e. for some a 2 M0 with a > 0, we have that a bi a for i D 1; : : : ; n. To go on to find a solution in M0n requires a separate argument, which I shall not go into in these notes. The method is fairly routine and appeared several years before the eventual proof of the model completeness of Texp (see [15]). The case n D 1 of both arguments is reasonably straightforward and serves as a good introduction to the general case: Exercise 2. Let M0 D hM0 ; :::i and M D hM; :::i be models of Texp with M0 M. Use the two step method discussed above to show that every zero in M of a one variable (nonzero) quasipoly
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