Which set existence axioms are needed to prove the separable Hahn-Banach theorem?
type: Item
instance of: scholarly article
title: Which set existence axioms are needed to prove the separable Hahn-Banach theorem?
zbMATH Open document ID: 0615.03044
DOI: 10.1016/0168-0072(86)90066-7
author: Douglas K. Brown
author: Stephen G. Simpson
published in: Annals of Pure and Applied Logic
publication date: 1986-01-01T00:00:00Z
review text: The authors work in the context of weak subsystems of second order arithmetic. \(RCA_ 0\) is the system with \(\Delta^ 0_ 1\) comprehension and \(\Sigma^ 0_ 1\) induction. \(WKL_ 0\) is \(RCA_ 0\) plus weak König's lemma for trees of finite sequences of 0's and 1's. It is shown that the Hahn-Banach theorem for separable Banach spaces is equivalent to \(WKL_ 0\) over \(RCA_ 0\). To prove the separable Hahn- Banach theorem within \(WKL_ 0\), one has to refine the usual proof which uses the \(\omega\)-times iterated arithmetical comprehension axiom. For the other direction, the authors make use of ideas of Bishop, Metakides, Nerode and Shore. The paper is well-written and gives just the right amount of detail. However, in a number of places the reader is referred to the PhD Thesis of the first author and a forthcoming book of the second author.
Mathematics Subject Classification ID: 03F35
zbMATH DE Number: 3995668
zbMATH Keywords: reverse mathematics
zbMATH Keywords: weak subsystems of second order arithmetic
zbMATH Keywords: weak König's lemma
zbMATH Keywords: Hahn-Banach theorem for separable Banach spaces
zbMATH Keywords: arithmetical comprehension axiom
reviewed by: Helmut Schwichtenberg
MaRDI profile type: Publication
full work available at URL: 0168-0072(86)90066-7
OpenAlex ID: W2087408122
Wikidata QID: Q56049305
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description: scientific article; zbMATH DE number 3995668