FROM BOUNDED ARITHMETIC TO SECOND ORDER ARITHMETIC VIA AUTOMORPHISMS
Ali Enayat · 2006
In this paper we examine the relationship between automorphisms of models of I∆ 0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA 0 (arithmetical comprehension schema with restricted induction), and Z 2 (second order arithmetic).For example, we establish the following characterization of P A by proving a "reversal" of a theorem of Gaifman:Theorem.The following are equivalent for completions T of I∆ 0 : (a) T P A; (b) Some model M = (M, • • •) of T has a proper end extension N which satisfies I∆ 0 and for some automorphism j of N, M is precisely the fixed point set of j.Our results also shed light on the metamathematics of the Quine-Jensen system N F U of set theory with a universal set.