Avoiding logical strength in real analysis

TL;DR

This paper shows that many core results in one-dimensional real analysis can be developed in weaker logical systems without the need for traditional convergence rates, challenging previous assumptions about their logical strength.

Contribution

It demonstrates that classical theorems like Bolzano-Weierstrass and Heine-Borel can be proved in conservative theories over RCA_0 using slow Cauchy sequences, reducing the perceived logical strength.

Findings

Most real analysis results can be developed without convergence rates.

Theorems like Bolzano-Weierstrass and Heine-Borel are provable in weaker systems.

The division between analytical and combinatorial principles in reverse mathematics may be less strict.

Abstract

In reverse mathematics, real numbers are traditionally represented by Cauchy sequences with a given rate of convergence. We work without rates and speak of slow Cauchy sequences. It turns out that almost all one-dimensional real analysis from the reverse mathematics book by Simpson can then be developed in theories that are conservative over $\mathsf{RCA}_0$. Specifically, we obtain clusters of equivalences with the infinite pigeonhole principle and the strong cohesive principle. The second cluster includes results like the Bolzano-Weierstrass and Arzel\`a-Ascoli theorems, which are traditionally associated with the stronger axiom of arithmetical comprehension, but also the Heine-Borel theorem, which is normally separated from these principles. This suggests two things: In elementary analysis, one can avoid logical strength to an extent that the traditional picture seems to forbid. And the division of the so-called reverse mathematics zoo into analytical and combinatorial principles may be less rigid than previously assumed.