Model Completeness of O-Minimal Fields With Convex Valuations

Abstract

We let R be an o-minimal expansion of a field, V a convex subring, and (R 0 , V 0 ) an elementary substructure of (R, V ). Our main result is that (R, V ) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R, provided that k R (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k with structure induced from (R, V ). We also give a criterion for a superstructure of (R, V ) being an elementary extension of (R, V ).

Measuring definable sets in o-minimal fields

“…From now on we shall assume that R is ω-saturated, in order to have k = R. This is no loss of generality: By Theorem 3.3 in [4], for any elementary…”

The Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field

“…We now define a d-dimensional measure ν d on the d-thin definable subsets of O n such that ν d X > 0 whenever dim X ≥ d. On d-fat sets one can define a d-dimensional measure as Fornasiero, Vasquez do in [3]. While the proofs in [3] work only in a sufficiently saturated o-minimal expansion of a real closed field, using Theorem 3.3 on p.244 in [2], the results from [3] can be transferred to the general case.…”

Quantifier elimination for o-minimal structures expanded by a valuational cut

“…A particularly well-behaved class of such structures are the T -convex structures, as introduced by van den Dries and Lewenberg in [6]. But even when one relaxes the T-convexity condition to the strictly weaker condition of having an o-minimal residue field, one maintains some desirable properties such as first-order axiomatizability (see Maříková [12]), and model-completeness of (R, C) in an expansion of L by a small number of constants (see Ealy, Maříková [8]). Later, the second author noticed (in unpublished notes) that the proof in [8] can rather easily be extended to yield quantifier elimination.…”

“…But even when one relaxes the T-convexity condition to the strictly weaker condition of having an o-minimal residue field, one maintains some desirable properties such as first-order axiomatizability (see Maříková [12]), and model-completeness of (R, C) in an expansion of L by a small number of constants (see Ealy, Maříková [8]). Later, the second author noticed (in unpublished notes) that the proof in [8] can rather easily be extended to yield quantifier elimination. The question remained whether the assumption of o-minimality of the residue field is necessary.…”

“…In [8], to prove model completeness, one first establishes Theorem 3.3, p. 244, a criterion for elementary extensions of (R, C). While [8] assumes that R expands a field, C is a convex subring, in this paper we show that these hypotheses may be replaced by the hypotheses that R expands a group and C defines a valuational cut.…”