The cohesive principle and the Bolzano-Weierstrass principle

作者：Kreuzer Alexander P^*

来源：Mathematical Logic Quarterly, 2011, 57(3): 292-298.

DOI：10.1002/malq.201010008

• Bolzano-Weierstrass principle
• cohesive principle
• sequential compactness

The aim of this paper is to determine the logical and computational strength of instances of the BolzanoWeierstrass principle (BW) and a weak variant of it.
We show that BW is instance-wise equivalent to the weak Konig'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).

• 出版日期2011-6