After introducing basic notation and results in chapter one, we begin studying the model theory of the Peano axioms, P, proper in the second chapter where we give a proof of Rabin's theorem : that P is not axiomatizable by any consistent set of [sigma]n sentences for any n[epsilon][omega], and also answer a question of Gaifman raised in Another problem, from the same article, is partially answered in chapter three, where we show every countable nonstandard model, M, 'of P has an elementary equivalent end extension solving a Diophantine equation with coefficients in M, that was not solvable in M. In chapter four we investigate substructures of countable nonstandard models of P, and show that every such model M, contains 2 substructures all isomorphic to M. Other related results are also proved. Chapter five contains theorems on indescernibles and omitting certain types in models of P. Chapter six is concerned with the following problem the set), of elementary substructures of M, is lattice ordered by inclusion. Which lattices are of the form for some We show that the pentagon lattice is of this form (answering a question suggested in [7] p. 280)and produce a class of nonmodular lattices all of whose members are not of the form for any M = N, the standard model of P. Elementary cofinal extensions of models of P are also investigated in this chapter. Finally, chapter seven concludes the thesis by posing some open problems suggested by the preceding text.
