2015/11/15 by Sam Sanders, Sanders, Sam
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.1511.04679
arXiv admin note: substantial text overlap with arXiv:1508.07434
arxiv created 2015/11/15 · arxiv updated 2015/11/17
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.