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We introduce a new notion of computable function on R and prove some basic properties. We give two applications, first a short proof of Yoshinaga’s theorem that periods are elementary (they are actually lower elementary). We also show that the lower elementary complex numbers form an algebraically closed field closed under exponentiation and some other(More)
Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ’large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow(More)
We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group(More)