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A model-theoretic version of the complement theorem : applications

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 4

355-361

The paper treats of some consequences of the model-theoretic version of Gabrielov's complement theorem from [11], which asserts that the theories T[sub an] (introduced in [11] and T'[sub an] (defined herein) are model-complete. The theory T'[sub an] is a universal modification of T[sub an] in the language L'[sub an] of ordered rings expanded by the symbols of restricted analytic functions, arithmetic roots and multiplicative inverse l/x. We give a short proof of the curve selecting lemma, and next we demonstrate how quantifier elimination, within the structure R[sub an] expanded by multiplicative inverse 1/x (a result due to Denef-van den Dries [4], can be obtained from the complement theorem through a general method of logic. Also presented is an application to definability problems ; namely, a piecewise description of a subanalytic function by restricted analytic functions, arithmetic roots and l/x.

A model-theoretic version of the complement theorem

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 4

345-353

This paper deals with an axiomatic theory T[sub an] and the expansion R[sub an] of the ordered field of reals, formed by attaching the restricted analytic functions. We show that the theory T[sub an] is model-complete, which may be regarded as a version of Gabrielov's complement theorem. Our proof is based on Robinson's test and it does not involve a partition technique. An immediate corollary is that T[sub an] coincides with the semantic theory Th(R[sub an]) of all sentences true in the structure R[sub an].