Weak theories of nonstandard arithmetic and analysis

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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H. Jerome Keisler (2010). Nonstandard Arithmetic and Recursive Comprehension. Annals of Pure and Applied Logic 161 (8):1047-1062.
Sam Sanders (2010). More Infinity for a Better Finitism. Annals of Pure and Applied Logic 161 (12):1525-1540.

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