Chris Miller : Expansions of o-minimal structures by fast sequences

Posted by Matthias Aschenbrenner , part of the Logic Seminar.

At

Aug. 23, 2005, 4 p.m.

In

SEO 512

Abstract

Let $\mathfrak R$ be an o-minimal expansion of $(\mathbb R,<,+)$ and $(\phi_k)_{k\in\mathbb N}$ be a sequence of positive real numbers such that $\lim_{k\to+\infty}f(\phi_k)/\phi_{k+1}=0$ for every $f \colon\mathbb R\to \mathbb R$ definable in $\mathfrak R$. (Such sequences always exist under some reasonable extra assumptions on $\mathfrak R$, in particular, if $\mathfrak R$ is exponentially bounded or if the language is countable.) Then $\bigl(\mathfrak R, (S)\bigr)$ is d-minimal, where $S$ ranges over all subsets of cartesian powers of the range of $\phi$. (Joint work with Harvey Friedman. Published in JSL 70.)