Saturation and \(\Sigma_2\)-transfer for ERNA.

*(English)*Zbl 1184.03071The paper continues the development of a weak theory of nonstandard analysis called ERNA (Elementary Recursive Nonstandard Analysis), introduced by P. Suppes and R. Sommer [Notas de la Sociedad Matemática de Chile 15, 73–95 (1995)], which was also considered in a previous paper by the present authors [J. Symb. Log. 73, No. 2, 689–710 (2008; Zbl 1141.03032)]. In this paper, the authors consider a version of the \(\Sigma^0_2\)-transfer principle (\(\Sigma_2\)-TRANS), and show that the consistency of ERNA + \(\Sigma_2\)-TRANS is provable in the primitive recursive arithmetic (PRA). Next, they introduce an (unrestricted complexity) saturation principle (SAT), whose consistency with ERNA is also claimed to be provable in PRA. They also discuss a generalization of the transfer principle to certain nonstandard formulas, and interpretations of classical subsystems of arithmetic based on the transfer principle.

I observe that many results in the paper look suspect, to put it mildly. For example, the authors simultaneously claim that on the one hand, ERNA + \(\Sigma_2\)-TRANS interprets \(\text{I}\Sigma_1\) and ERNA + \(\Pi_1\)-TRANS interprets \(\text{B}\Sigma_2\), and on the other hand that the consistency of these two theories is provable in PRA, contradicting Gödel’s incompleteness theorem in both cases.

I observe that many results in the paper look suspect, to put it mildly. For example, the authors simultaneously claim that on the one hand, ERNA + \(\Sigma_2\)-TRANS interprets \(\text{I}\Sigma_1\) and ERNA + \(\Pi_1\)-TRANS interprets \(\text{B}\Sigma_2\), and on the other hand that the consistency of these two theories is provable in PRA, contradicting Gödel’s incompleteness theorem in both cases.

Reviewer: Emil Jeřábek (Praha)

##### Keywords:

nonstandard methods; transfer principle; saturation principle; nonstandard analysis; primitive recursive arithmetic
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\textit{C. Impens} and \textit{S. Sanders}, J. Symb. Log. 74, No. 3, 901--913 (2009; Zbl 1184.03071)

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