Unique solutions

Mathematical Logic Quarterly 52 (6):534-539 (2006)

Peter Schuster
University of Leeds
It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non-uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges
Keywords reverse mathematics  constructive mathematics  unique existence  continuity  fan theorem  Metric spaces  compactness
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DOI 10.1002/malq.200610012
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References found in this work BETA

Equivalents of the (Weak) Fan Theorem.Iris Loeb - 2005 - Annals of Pure and Applied Logic 132 (1):51-66.
Weak König's Lemma Implies Brouwer's Fan Theorem: A Direct Proof.Hajime Ishihara - 2006 - Notre Dame Journal of Formal Logic 47 (2):249-252.
Continuity Properties in Constructive Mathematics.Hajime Ishihara - 1992 - Journal of Symbolic Logic 57 (2):557-565.

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Citations of this work BETA

Corrigendum to “Unique Solutions”.Peter Schuster - 2007 - Mathematical Logic Quarterly 53 (2):214-214.

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