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

Article by Sanders, Sam
  • Published in 2015
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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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@article{Sanders2015,
	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.},
	author = {Sanders, Sam},
	month = {nov},
	title = {The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics},
	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},
	collections = {Attention-grabbing titles}
}