# Tameness and metric dimensions in expansions of the real field

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Christopher Miller

created by daniele on 27 May 2015

3 jun 2015
-- 14:30

Sala Seminari, DM, Pisa

Seminari di o-minimalità, Pisa

**Abstract.**

It is long known that any expansion, M, of the field of real numbers that defines N (the set of all natural numbers) also defines every real Borel set (of any arity), hence also every real projective (in the sense of descriptive set theory) set.

Thus, one can easily ask questions about the definable sets of M that turn out to be independent of ZFC (e.g., whether every definable set is Lebesgue measurable).
This leads naturally to wondering what can be said about its definable sets if M does not define N.

Philipp Hieronymi and I have recently obtained a result that can be stated loosely as: M avoids defining N if and only if all metric dimensions commonly encountered in geometric measure theory, fractal geometry and analysis on metric spaces coincide with topological dimension on all images of closed definable sets under definable continuous maps.