The strength of nonstandard methods in arithmetic

Journal of Symbolic Logic 49 (4):1039-1058 (1984)
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
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P. V. Andreev & E. I. Gordon (2006). A Theory of Hyperfinite Sets. Annals of Pure and Applied Logic 143 (1):3-19.
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