Weak theories of nonstandard arithmetic and analysis

Abstract
A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.
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A Functional Interpretation for Nonstandard Arithmetic.Benno van den Berg, Eyvind Briseid & Pavol Safarik - 2012 - Annals of Pure and Applied Logic 163 (12):1962-1994.
More Infinity for a Better Finitism.Sam Sanders - 2010 - Annals of Pure and Applied Logic 161 (12):1525-1540.
Nonstandard Arithmetic and Recursive Comprehension.H. Jerome Keisler - 2010 - Annals of Pure and Applied Logic 161 (8):1047-1062.

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