1709.03150Sep 10, 2017A previous version of this paper was disseminated under the title "On continuous functions definable in expansions of the ordered real additive group''

A tetrachotomy for expansions of the real ordered additive group

Philipp Hieronymi, Erik Walsberg

Logic

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Abstract

Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval $I\subseteq \mathbb{R}$

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, or

\mathcal{R}

is a reduct of an ordered vector space. We say

\mathcal{R}

is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of

(\mathbb{R},<,+)

. In particular, we show that for expansions that do not define dense

ω

-orders (we call these type A expansions), an appropriate version of Zilber's principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function