The strength of nonstandard methods in arithmetic

Journal of Symbolic Logic 49 (4):1039-1058 (1984)

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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DOI 10.2307/2274260
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Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Mathematics in the Alternative Set Theory.[author unknown] - 1984 - Journal of Symbolic Logic 49 (4):1423-1424.

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Nonstandard Arithmetic and Reverse Mathematics.H. Jerome Keisler - 2006 - Bulletin of Symbolic Logic 12 (1):100-125.
A Theory of Hyperfinite Sets.P. V. Andreev & E. I. Gordon - 2006 - Annals of Pure and Applied Logic 143 (1):3-19.
Hyperfinite Models of Adapted Probability Logic.H. Jerome Keisler - 1986 - Annals of Pure and Applied Logic 31 (1):71-86.

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