In a recent paper  H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem , and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of .
Firstly, it is shown in  that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin  stating the P is not implied by any consistent set of ∑n sentences for any n.
Secondly, we partially answer a question posed on p. 129 of  by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin .