The cohesive principle and the Bolzano-Weierstra{\ss} principle

TL;DR

This paper analyzes the logical and computational strength of the Bolzano-Weierstra{ extss} principle and its weak variant, establishing their equivalences with known principles and classifying their computational power.

Contribution

It establishes the equivalence of BW with weak K"onig's lemma for $oldsymbol{ m ext{ extSigma}^0_1}$-trees and classifies the weak BW as strictly weaker than BW, with limited computational strength.

Findings

BW is equivalent to weak K"onig's lemma for $ ext{ extSigma}^0_1$-trees

BW_weak is equivalent to the strong cohesive principle

BW_weak does not solve the halting problem and has primitive recursive growth

Abstract

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of it. We show that BW is instance-wise equivalent to the weak K\"onig's lemma for $\Sigma^0_1$-trees ($\Sigma^0_1$-WKL). This means that from every bounded sequence of reals one can compute an infinite $\Sigma^0_1$-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d >> 0' are exactly those containing an accumulation point for all bounded computable sequences. Let BW_weak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence (a sequence converging but not necessarily fast). We show that BW_weak is instance-wise equivalent to the (strong) cohesive principle (StCOH) and - using this - obtain a classification of the computational and logical strength of BW_weak. Especially we show that BW_weak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BW_weak.