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The cohesive principle and the Bolzano-Weierstraß principle

✍ Scribed by Alexander P. Kreuzer

Publisher: John Wiley and Sons
Year: 2011
Tongue: English
Weight: 97 KB
Volume: 57
Category: Article
ISSN: 0044-3050
DOI: 10.1002/malq.201010008

No coin nor oath required. For personal study only.

✦ Synopsis

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle (BW) and a weak variant of it.

We show that BW is instance-wise equivalent to the weak König's lemma for Σ 0 1 -trees Σ 0 1 -WKL . This means that from every bounded sequence of reals one can compute an infinite Σ 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 .

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