Minima of initial segments of infinite sequences of reals

✍ Scribed by Jeffry L. Hirst

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 90 KB
• Volume: 50
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200310075

✦ Synopsis

Abstract

Suppose that 〈x~k~〉~k∈ℕ~ is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA~0~ suffices to prove the existence of a sequence of reals 〈u~k~〉~k∈ℕ~ such that for each k, u~k~ is the minimum of {x~0~, x~1~, …, x~k~}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈x~k~〉~k∈ℕ~, the stronger subsystem WKL~0~ is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)