A note on Bar Induction in Constructive Set Theory

Mathematical Logic Quarterly 52 (3):253-258 (2006)

Michael Rathjen
University of Leeds
Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF
Keywords Bar Induction  Constructive set theory  proof‐theoretic strength  Brouwerian principles
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DOI 10.1002/malq.200510030
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Constructive Set Theory.John Myhill - 1975 - Journal of Symbolic Logic 40 (3):347-382.
Inaccessible Set Axioms May Have Little Consistency Strength.L. Crosilla & M. Rathjen - 2002 - Annals of Pure and Applied Logic 115 (1-3):33-70.

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