The complexity of classification problems for models of arithmetic

Samuel Coskey, Roman Kossak

Published 2009-08-12, updated 2010-10-27Version 2

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.