A note on Bar Induction in Constructive Set Theory

Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive ZermeloFraenkel 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. Key words: Constructive set theory, Brouwerian principles, Bar Induction, prooftheoretic strength

• Type: Journal
• Year: 2006
• Where: MLQ
• Authors: Michael Rathjen
• Keywords: Bar Induction, Decidable Bar Induction, MLQ 2006, ZermeloFraenkel Set Theory