### Principles of bar induction and continuity on Baire space

Tatsuji Kawai
2019 Journal of Logic and Analysis
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 Brouweroperation. Working in Bishop constructive mathematics, we show that the above principle is equivalent to a
more » ... n 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 Π 0 1 bar induction in general implies LLPO (the lesser limited principle of omniscience). This, together with the fact that the Σ 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. a well-known recursive counterexample (see Troelstra and van Dalen [10, Chapter 4, Section 7.6]). Here, the fan theorem is a statement saying that every bar of Cantor space is uniform (see Section 2 for terminology). The connection between UC and the fan theorem is well studied in constructive reverse mathematics (Ishihara [6]). It is well known that the fan theorem is equivalent to compactness of Cantor space [10, Chapter 4, Section 6], and hence it implies UC. Josef Berger [2] showed that a weaker version of UC is equivalent to the decidable fan theorem (see also Remark 5.4). In another paper [3], he also introduced a variant of fan theorem, called c-FT, and showed that it is equivalent to UC. In this paper, we establish analogous correspondence between several notions of continuity on Baire space N N and a variety of bar induction. Our focus is on the relation between various versions of bar induction and statements similar to UC, but we consider functions on Baire space instead of Cantor space and replace uniform continuity with a suitable notion of continuity on Baire space. More precisely, we consider a function from N N to N induced by a Brouwer-operation (Kreisel and Troelstra [9, Section 3]) to be a fundamental notion of continuity on Baire space. The notion can be considered as a natural generalisation of that of uniform continuity on Cantor space to the setting of Baire space, since it becomes equivalent to uniform continuity when restricted to Cantor space (see Proposition 3.2). We now summarise our main contributions. First, we formulate a continuity principle for Baire space called the principle of Brouwer continuity ( BC ), based on the notion of Brouwer-operation. The principle BC states that every pointwise continuous function from Baire space to the set of natural numbers is induced by a Brouwer-operation. Then, we introduce a variant of bar induction, called the continuous bar induction (c-BI), and show that c-BI is equivalent to BC . Moreover, we characterise the other versions of bar induction, the monotone bar induction and the decidable bar induction, by a stronger and a weaker version of BC by varying the strength of the premise of BC . Finally, we show that the Π 0 1 bar induction (of which c-BI is an instance) in general implies the non-constructive principle LLPO (the lesser limited principle of omniscience), and thus intuitionistically unacceptable. 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.