  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.
