Notre Dame Journal of Formal Logic

Refining the Taming of the Reverse Mathematics Zoo

Sam Sanders

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Abstract

Reverse mathematics is a program in the foundations of mathematics. It provides an elegant classification in which the majority of theorems of ordinary mathematics fall into only five categories, based on the “big five” logical systems. Recently, a lot of effort has been directed toward finding exceptional theorems, that is, those which fall outside the big five. The so-called reverse mathematics zoo is a collection of such exceptional theorems (and their relations). It was previously shown that a number of uniform versions of the zoo theorems, that is, where a functional computes the objects stated to exist, fall in the third big five category, arithmetical comprehension, inside Kohlenbach’s higher-order reverse mathematics. In this paper, we extend and refine these previous results. In particular, we establish analogous results for recent additions to the reverse mathematics zoo, thus establishing that the latter disappear at the uniform level. Furthermore, we show that the aforementioned equivalences can be proved using only intuitionistic logic. Perhaps most surprisingly, these explicit equivalences are extracted from nonstandard equivalences in Nelson’s internal set theory, and we show that the nonstandard equivalence can be recovered from the explicit ones. Finally, the following zoo theorems are studied in this paper: Π10G (existence of uniformly Π10-generics), FIP (finite intersection principle), 1-GEN (existence of 1-generics), OPT (omitting partial types principle), AMT (atomic model theorem), SADS (stable ascending or descending sequence), AST (atomic model theorem with subenumerable types), NCS (existence of noncomputable sets), and KPT (Kleene–Post theorem that there exist Turing incomparable sets).

Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 4 (2018), 579-597.

Dates
Received: 12 October 2015
Accepted: 12 June 2016
First available in Project Euclid: 12 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1539309631

Digital Object Identifier
doi:10.1215/00294527-2018-0015

Mathematical Reviews number (MathSciNet)
MR3871902

Zentralblatt MATH identifier
06996545

Subjects
Primary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
Secondary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05]

Keywords
higher-order reverse mathematics reverse mathematics zoo nonstandard analysis

Citation

Sanders, Sam. Refining the Taming of the Reverse Mathematics Zoo. Notre Dame J. Formal Logic 59 (2018), no. 4, 579--597. doi:10.1215/00294527-2018-0015. https://projecteuclid.org/euclid.ndjfl/1539309631


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