Rathjen, M (2016) Remarks on Barr's theorem : Proofs in geometric theories. In: Probst, D and Schuster, P, (eds.) Concepts of Proof in Mathematics, Philosophy, and Computer Science. Walter de Gruyter , pp. 347-373. ISBN 978-1-5015-0262-0
Abstract
A theorem, usually attributed to Barr, yields that (A) geometric implications deduced in classical L∞ω logic from geometric theories also have intuitionistic proofs. Barr’s theorem is of a topos-theoretic nature and its proof is non-constructive. In the literature one also finds mysterious comments about the capacity of this theorem to remove the axiom of choice from derivations. This article investigates the proof-theoretic side of Barr’s theorem and also aims to shed some light on the axiom of choice part. More concretely, a constructive proof of the Hauptsatz for L∞ω is given and is put to use to arrive at a simple proof of (A) that is formalizable in constructive set theory and Martin-L¨of type theory.
Metadata
Item Type: | Book Section |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 30 Sep 2016 14:27 |
Last Modified: | 04 Nov 2016 07:29 |
Status: | Published |
Publisher: | Walter de Gruyter |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:96231 |