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2009 How Incomputable Is the Separable Hahn-Banach Theorem?
Guido Gherardi , Alberto Marcone
Notre Dame J. Formal Logic 50(4): 393-425 (2009). DOI: 10.1215/00294527-2009-018


We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.


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Guido Gherardi . Alberto Marcone . "How Incomputable Is the Separable Hahn-Banach Theorem?." Notre Dame J. Formal Logic 50 (4) 393 - 425, 2009.


Published: 2009
First available in Project Euclid: 11 February 2010

zbMATH: 1223.03052
MathSciNet: MR2598871
Digital Object Identifier: 10.1215/00294527-2009-018

Primary: 03F60
Secondary: 03B30 , 46A22‎ , 46S30

Keywords: Computable analysis , Hahn-Banach extension theorem , multivalued functions , reverse mathematics , weak Konig's lemma

Rights: Copyright © 2009 University of Notre Dame


Vol.50 • No. 4 • 2009
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