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Two constructive embedding‐extension theorems with applications to continuity principles and to Banach‐Mazur computability

Andrej Bauer & Alex Simpson

Mathematical Logic Quarterly 50 (4‐5):351-369 (2004)

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Realizability models refuting Ishiharaʼs boundedness principle.Peter Lietz & Thomas Streicher - 2012 - Annals of Pure and Applied Logic 163 (12):1803-1807.details Ishiharaʼs boundedness principleBD-N was introduced in Ishihara [5] and has turned out to be most useful for constructive analysis, see e.g. Ishihara [6]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from NN to N is sequentially continuous and in which the axiom of choice from NN to N (...) Download Export citation Bookmark 1 citation
On local non‐compactness in recursive mathematics.Jakob G. Simonsen - 2006 - Mathematical Logic Quarterly 52 (4):323-330.details A metric space is said to be locally non-compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non-compact iff it is without isolated points.The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence (...) Download Export citation Bookmark

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