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arXiv · 1511.04679

The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics

Sam Sanders
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Abstract

The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis pertaining to Reverse Mathematics (RM). In particular, we shall establish RM-equivalences between theorems from Nonstandard Analysis in a fragment of Nelson's internal set theory. We then extract primitive recursive terms from Goedel's system T (not involving Nonstandard Analysis) from the proofs of the aforementioned nonstandard equivalences. The resulting terms turn out to be witnesses for effective1 equivalences in Kohlenbach's higher-order RM. In other words, from an RM-equivalence in Nonstandard Analysis, we can extract the associated effective higher-order RM-equivalence which does not involve Nonstandard Analysis anymore. Finally, we show that certain effective equivalences in turn give rise to the original nonstandard theorems from which they were derived.

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Sam Sanders. 2015-11-15. The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics. https://arxiv.org/abs/1511.04679

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