Interpreting the Monadic Second Order Theory of One Successor in Expansions of the Real Line

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 NIP or even NTP2. We use this to deduce the first general results about definable sets in NTP2 expansions of (R, <,+). The goal of this paper is to bring together the study of combinatorial tameness properties of first order structures initiated by Morley and Shelah (neostability) and the study of geometric tameness properties of expansions of the real line (R, <) championed by Miller (tame geometry, see [Mil05]). Let B be the two-sorted (first order) structure (P(N),N,∈,+1) where P(N) is the power set of N and +1 is the successor function on N. The theory of this structure is essentially the monadic second order theory of (N,+1). While Büchi showed in his landmark paper [Büc62] that B admits quantifier elimination in a suitable language and its theory is decidable, B obviously does not enjoy any Shelah-style combinatorial tameness properties, such as NIP or NTP2 (see e.g. Simon [Sim15] for definitions). Therefore any structure that defines an isomorphic copy of B, can not satisfy any of those properties, and has to be considered complicated or wild in this framework of combinatorial tameness. Here we study the consequences of the non-definability of a copy of B in an expansion of (R, <) on the geometric tameness of definable sets in this expansion. Our results are new even when the assumption “does not define an isomorphic copy of B” is replaced by one of the stronger assumptions “has NTP2” or even “has NIP”. Therefore these are arguably the very first general results about NTP2 expansions of (R, <,+). Throughout definable will always mean definable with parameters. We will say that a structure defines B if it defines an isomorphic copy of B. The main technical result of this paper is as follows. Theorem A. Let R = (R, <,D,≺) where D ⊆ R is dense in some open interval and ≺ is an order on D with order type ω. Then R defines B. Date: November 22, 2016. 2010 Mathematics Subject Classification. Primary 03C45 Secondary 03C64, 03D05, 28A80, 54F45.