## Notre Dame Journal of Formal Logic

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

### Refining the Taming of the Reverse Mathematics Zoo

#### 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: ${\Pi}_{1}^{0}G$ (existence of uniformly ${\Pi}_{1}^{0}$-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