## ERNA and Friedman’s reverse mathematics.(English)Zbl 1231.03059

The paper develops the weak theory of nonstandard analysis ERNA by R. Sommer and P. Suppes [“Finite models of elementary recursive nonstandard analysis”, Notas Soc. Mat. Chile 15, 73–95 (1996)] and its extension by a $$\Pi_1$$-transfer principle introduced by C. Impens and S. Sanders [“Transfer and a supremum principle for ERNA”, J. Symb. Log. 73, No. 2, 689–710 (2008; Zbl 1141.03032)]. The author proves that the $$\Pi_1$$-transfer principle can be generalized to a certain class of formulas involving nonstandard parameters (bar transfer). The paper goes on to develop basic notions from real analysis (continuity, differentiation, Riemann integration). The main theorem is a reverse-mathematics result for $$\mathrm{ERNA}+\Pi_1\mathrm{-TRANS}$$: it is shown that $$\Pi_1$$-transfer is equivalent over ERNA to suitable variants of various statements from analysis, such as the Cauchy completeness principle, the Weierstrass approximation theorem, the fundamental theorem of calculus, or the Peano existence theorem for ODE. The author notes that these equivalents are generally versions of principles equivalent to $$\mathrm{WKL}_0$$ in standard reverse mathematics, with some equalities weakened so that they only hold up to infinitesimals.

### MSC:

 03H05 Nonstandard models in mathematics 03F35 Second- and higher-order arithmetic and fragments 26E35 Nonstandard analysis

Zbl 1141.03032
Full Text:

### References:

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