Submission date: 10 September 2017

Abstract:

Let R be an expansion of the ordered real additive group. Then one of the following holds: either every continuous function [0,1] → R definable in R is C^2 on an open dense subset of [0,1], or every C^2 function [0,1] → R definable in R is affine, or every continuous function [0,1] → R is definable in R.
If R is NTP_{2} or more generally does not interpret a model of the monadic second order theory of one successor, the first case holds. It is due to Marker, Peterzil, and Pillay that whenever R defines a C^2 function [0,1] → R that is not affine, it also defines an ordered field on some open interval whose ordering coincides with the usual ordering on R. Assuming R does not interpret second-order arithmetic, we show that the last statement holds when C^2 is replaced by C^1.

Mathematics Subject Classification: Primary 03C64 Secondary 03C40, 03D05, 03E15, 26A21, 54F45

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Full text arXiv 1709.03150: pdf, ps.