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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

Citations:

Zbl 1141.03032
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References:

[1] Techniques of constructive analysis (2006) · Zbl 1107.03065
[2] Notas de la Sociedad Mathematica de Chile 15 pp 73– (1996)
[3] Constructive analysis 279 (1985) · Zbl 0656.03042
[4] Reverse mathematics 2001 21 pp 19– (2005) · Zbl 1097.03053
[5] DOI: 10.1016/j.apal.2010.06.003 · Zbl 1223.03055
[6] DOI: 10.1007/s00153-007-0050-6 · Zbl 1114.03047
[7] DOI: 10.2178/bsl/1174668218 · Zbl 1129.03039
[8] DOI: 10.2178/bsl/1140640945 · Zbl 1101.03040
[9] Philosophia Scientiae (Cahier Spécial) 6 pp 43– (2006)
[10] From sets and types to topology and analysis 48 pp 245– (2005)
[11] Saturation and {\(\Sigma\)}2-transfer for ERNA 74 pp 901– (2009)
[12] Transfer and a supremum principle for ERNA 73 pp 689– (2008) · Zbl 1141.03032
[13] DOI: 10.1007/978-3-211-49905-4_5
[14] An introduction to nonstandard real analysis 118 (1985) · Zbl 0583.26006
[15] Subsystems of second order arithmetic (2009) · Zbl 1181.03001
[16] Reverse mathematics 2001 21 (2005)
[17] Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? 49 pp 783– (1984) · Zbl 0584.03039
[18] Systems of second order arithmetic with restricted induction, I & II (abstracts) 41 pp 557– (1976)
[19] Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974) 1 pp 235– (1975)
[20] Real analysis with real applications (2002)
[21] Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof 60 pp 122– (1995) · Zbl 0833.03021
[22] DOI: 10.1016/S0049-237X(98)80016-5
[23] DOI: 10.1007/s00153-006-0017-z · Zbl 1112.03053
[24] Proceedings of the IXth Latin American Symposium on Mathematical Logic 38 pp 1– (1993)
[25] DOI: 10.1006/jmps.1997.1142 · Zbl 0911.03036
[26] Constructive functional analysis 28 (1979) · Zbl 0401.03027
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