From Bolzano-Weierstra{\ss} to Arzel\`a-Ascoli

TL;DR

This paper connects the Arzelà-Ascoli theorem with the Bolzano-Weierstraß principle, classifying their logical strengths and equivalences within reverse mathematics, and explores their variants and related principles.

Contribution

It establishes equivalences between the Arzelà-Ascoli theorem and Bolzano-Weierstraß principles within reverse mathematics, providing a classification of their logical strengths and variants.

Findings

AA is equivalent to BW over RCA_0

AA_weak is equivalent to BW_weak over WKL_0

AA_weak, BW_weak + WKL, and StCOH + WKL are equivalent over RCA_0

Abstract

We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of instances of the Arzel\`a-Ascoli theorem and a variant of it. Let AA be the statement that each equicontinuous sequence of functions f_n: [0,1] --> [0,1] contains a subsequence that converges uniformly with the rate 2^-k and let AA_weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that AA is instance-wise equivalent over RCA_0 to the Bolzano-Weierstra{\ss} principle BW and that AA_weak is instance-wise equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL and StCOH + WKL are equivalent.