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Mathematics > Logic

arXiv:1804.07952 (math)
[Submitted on 21 Apr 2018 (v1), last revised 9 May 2018 (this version, v2)]

Title:Some nonstandard equivalences in Reverse Mathematics

Authors:Sam Sanders
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Abstract: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 weak base theory recently introduced by van den Berg and the author. In particular, our base theories have the first-order strength of elementary function arithmetic, in contrast to the original version of this paper [22]. Our results combined with Simpson's criterion for naturalness suggest the controversial point that Nonstandard Analysis is actually mathematically natural.
Comments: 10 pages, original version published in LNCS, Proceedings of CiE18
Subjects: Logic (math.LO)
Cite as: arXiv:1804.07952 [math.LO]
  (or arXiv:1804.07952v2 [math.LO] for this version)
  https://doi.org/10.48550/arXiv.1804.07952

Submission history

From: Sam Sanders [view email]
[v1] Sat, 21 Apr 2018 12:06:24 UTC (28 KB)
[v2] Wed, 9 May 2018 07:51:16 UTC (65 KB)
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