The strength of nonstandard methods in arithmetic

Abstract

Abstract We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N ( x ) for an elementary initial segment, along with axiom schemes approximating ω 1 -saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.