How Incomputable Is the Separable Hahn-Banach Theorem?

Notre Dame Journal of Formal Logic 50 (4):393-425 (2009)
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Abstract

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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References found in this work

Effective Borel measurability and reducibility of functions.Vasco Brattka - 2005 - Mathematical Logic Quarterly 51 (1):19-44.
Fixed point theory in weak second-order arithmetic.Naoki Shioji & Kazuyuki Tanaka - 1990 - Annals of Pure and Applied Logic 47 (2):167-188.
Borel complexity and computability of the Hahn–Banach Theorem.Vasco Brattka - 2008 - Archive for Mathematical Logic 46 (7-8):547-564.
Computable metrization.Tanja Grubba, Matthias Schröder & Klaus Weihrauch - 2007 - Mathematical Logic Quarterly 53 (4‐5):381-395.

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