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

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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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- Sanders2015
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- article
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- 2015-11-19
- date_published
- 2015-11-01

### BibTeX entry

@article{Sanders2015, key = {Sanders2015}, type = {article}, title = {The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics}, author = {Sanders, Sam}, 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.}, comment = {}, date_added = {2015-11-19}, date_published = {2015-11-01}, urls = {http://arxiv.org/abs/1511.04679,http://arxiv.org/pdf/1511.04679v1}, collections = {Attention-grabbing titles,About proof}, month = {nov}, url = {http://arxiv.org/abs/1511.04679 http://arxiv.org/pdf/1511.04679v1}, year = 2015, archivePrefix = {arXiv}, eprint = {1511.04679}, primaryClass = {math.LO}, urldate = {2015-11-19} }