Interpreting the monadic second order theory of one successor in expansions of the real line

TL;DR

This paper establishes conditions under which certain expansions of the real line can define complex monadic second order structures, revealing that such expansions lack tameness properties like NIP or NTP2 and providing new insights into definable sets in these contexts.

Contribution

It provides the first general conditions for when expansions of the real line can define the monadic second order theory of one successor, showing they lack Shelah's tameness properties.

Findings

Expansions can define the standard model of monadic second order theory of one successor.

Such expansions do not satisfy NIP or NTP2 properties.

New results on the structure of definable sets in NTP2 expansions of the real line.

Abstract

We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties defined by Shelah, such as $\textrm{NIP}$ or even $\textrm{NTP}_2$. We use this to deduce the first general results about definable sets in $\textrm{NTP}_2$ expansions of $(\mathbb{R},<,+)$.