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.
Highlights
The uniform continuity principle (UC) is the following statement: UC Every pointwise continuous function F : {0, 1}N → N is uniformly continuous
We show that the Π01 bar induction in general implies the non-constructive principle LLPO, and intuitionistically unacceptable
Our main contribution is in introducing the bar induction continuous bar induction (c–BI) which is equivalent to BC and characterising the other versions of bar induction by similar principles of continuity
Summary
The uniform continuity principle (UC) is the following statement: UC Every pointwise continuous function F : {0, 1}N → N is uniformly continuous. The relation between several versions of bar induction and continuity axioms (namely strong and weak continuity for numbers, and bar continuity) has been extensively studied by Howard and Kreisel [5] and Kreisel and Troelstra [9]. Some of their results are recalled as corollaries of our work in Section 6 (Theorem 6.1). Our main contribution is in introducing the bar induction c–BI which is equivalent to BC and characterising the other versions of bar induction by similar principles of continuity In this way, the Journal of Logic & Analysis 11:FT3 (2019).
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