Unique existence, approximate solutions, and countable choice
An existential statement seems to admit of a constructive proof without countable choice only if the object to be constructed is uniquely determined, or is intended as an approximate solution of the problem in question. This conjecture is substantiated by re-examining some basic tools of mathematical analysis from a choice-free constructive point of view, starting from Dedekind cuts as an appropriate notion of real numbers. By establishing a fairly general version of the approximate intermediate value theorem, we also indicate that strong continuity is a practicable substitute for uniform continuity, where a mapping between metric spaces is called strongly continuous if two subsets of its domain are bounded away from each other whenever so are their images. (C) 2002 Elsevier B.V. All rights reserved.
countable choice intermediate values strong continuity constructive mathematics connectedness convexity