Published online by Cambridge University Press: 12 March 2014
Let = {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order -theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying
(i) K has no proper elementary substructures, and
(ii) whenever L ≻ K is a countable elementary extension there is and such that .
Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.