One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA is characterized up to isomorphism by two invariants: its first-order theory Th(^) and its standard system SSy {Jt). At present, there seems to be no indication of how to recover any information about Th {Jt) from Aut {Jt) with the exception of whether or not Th {Jt) is True Arithmetic. We define the notion of arithmetically saturated in Definition 1.7; however, a model Jt of PA is arithmetically saturated if and only if it is recursively saturated and the standard cut is a strong cut. The following is our main theorem. THEOREM. Suppose that Jtx and Jt2 are countable, arithmetically saturated models of PA such that Aut(^) s Aut {Jt2). Then SSy {Jtx) = SSy {Jt2). In the Theorem, it suffices to assume that Jt2 is just recursively saturated. For, as shown by Lascar [8], if Jtx and Jt2 are countable, recursively saturated models of PA

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Abstract. 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. 1.