The Biggest Five of Reverse Mathematics

Abstract: I provide an overview of joint work with Dag Normann on the higher-order Reverse Mathematics (RM for short) of the Big Five systems and the surprising limits of this enterprise ([3]).

The well-known Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the ‘Big Five’ of RM. The aim of this paper is to greatly extend the Big Five phenomenon, working in Kohlenbach’s higher-order RM ([1]).

In particular, we have established numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalisations and variations (involving e.g. the notions Baire 2 and cliquishness) of the aforementioned third-order theorems fall far outside of the Big Five. In particular, these slight generalisations and variations imply the principle NIN from [2], i.e. there is no injection from [0, 1] to N. We discuss a possible explanation for this phenomenon.

REFERENCES.

[1] Ulrich Kohlenbach, Higher order reverse mathematics, Reverse mathematics 2001, Lect. Notes Log., vol. 21, ASL, 2005, pp. 281–295.

[2] Dag Normann and Sam Sanders, On the uncountability of R, Journal of Symbolic Logic, DOI: doi.org/ 10.1017/jsl.2022.27 (2022), pp. 43.

[3] _________________________, The Biggest Five of Reverse Mathematics, Submitted, arxiv: arxiv.org/abs/2212.00489 (2023), pp. 39.

logic

Audience: researchers in the topic

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Online logic seminar

Series comments: Description: Seminar on all areas of logic

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