Principles of bar induction and continuity on Baire space

Abstract

Brouwer-operations, also known as inductively defined neighbourhood functions, provide a good notion of continuity on Baire space which naturally extends that of uniform continuity on Cantor space. In this paper, we introduce a continuity principle for Baire space which says that every pointwise continuous function from Baire space to the set of natural numbers is induced by a Brouwer-operation. Working in Bishop constructive mathematics, we show that the above principle is equivalent to a version of bar induction whose strength is between that of the monotone bar induction and the decidable bar induction. We also show that the monotone bar induction and the decidable bar induction can be characterised by similar principles of continuity. Moreover, we show that the $\Pi^{0}_{1}$ bar induction in general implies LLPO (the lesser limited principle of omniscience). This, together with a fact that the $\Sigma^{0}_{1}$ bar induction implies LPO (the limited principle of omniscience), shows that an intuitionistically acceptable form of bar induction requires the bar to be monotone.