Reverse Mathematics (RM) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson. The aim of RM is finding the minimal axioms needed to prove a theorem of ordinary (i.e. non-set theoretical) mathematics. In the majority of cases, one also obtains an equivalence between the theorem and its minimal axioms. This equivalence is established in a weak logical system called the base theory; four prominent axioms which boast lots of such equivalences are dubbed mathematically natural by Simpson. In this paper, we show that a number of axioms from Nonstandard Analysis are equivalent to theorems of ordinary mathematics not involving Nonstandard Analysis. These equivalences are proved in a base theory recently introduced by van den Berg and the author. Our results combined with Simpson’s criterion for naturalness suggest the controversial point that Nonstandard Analysis is actually mathematically natural.
CITATION STYLE
Sanders, S. (2018). Some nonstandard equivalences in reverse mathematics. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 10936 LNCS, pp. 365–375). Springer Verlag. https://doi.org/10.1007/978-3-319-94418-0_37
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