Constructive Connections between Anti-specker, Positivity, and Fan-theoretic Properties

Two weakenings of the anti-Specker property—a principle of some significance in constructive reverse mathematics—are introduced, examined, and in one case applied, within Bishop-style constructive mathematics. The weaker of these anti-Specker properties is shown to be equivalent to a very weak version of Brouwer’s fan theorem. This leads to a study of antitheses of various types of fan theorem—in particular, to new proofs of Diener’s theorem on the equivalence of some of these antitheses. In addition, the antithesis of the positivity principle for uniformly continuous functions on [0, 1] is shown to be equivalent to that of the fan theorem for detachable bars. Finally, a positivity principle for pointwise continuous functions is examined, partly in order to provide a neat application of the stronger of the two anti-Specker properties introduced early in the paper.