A continuity principle, a version of Baire’s theorem and a boundedness principle. (English) Zbl 1160.03041

This paper shows that \(\text{WC-N}'\), \(\text{BT}'+\lnot\text{LPO}\), and \(\text{BD-N}+\lnot\text{LPO}\) are constructively equivalent. The axiom \(\text{WC-N}'\) is a weakening of Brouwer’s scheme of weak continuity for numbers, \(\text{BT}'\) is a restricted version of Baire’s theorem, \(\text{LPO}\) is the limited principle of omniscience, and \(\text{BD-N}\) is the assertion that every enumerable pseudobounded set of natural numbers is bounded. Even though the authors work in the framework of Bishop’s constructive mathematics, they specify to which extent they use countable choice. Their result implies that \(\text{BT}'\) is valid in intuitionistic, classical, and constructive recursive mathematics.


03F60 Constructive and recursive analysis
03F65 Other constructive mathematics
03B30 Foundations of classical theories (including reverse mathematics)
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54E52 Baire category, Baire spaces
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