Consider a family S of restricted functions coming from a suitable quasianalytic class, and let RS denote the expansion of the real field by the functions in S. I will discuss an adaptation of the Rolin-Speissegger-Wilkie [1] model completeness construction which can be used to characterize when the theory of RS is decidable. This characterization is robust enough to allow the construction of examples of S, in a rather artificial manner, so that S contains transcendental functions and RS has a decidable theory. The title of the talk comes from the fact that the properties it lists characterize up to definable equivalence the types of structures, RS, discussed in the talk.

References

1. J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and ominimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777 (electronic).