Monotone convergence theorems equivalent to Markov's principle

Abstract

The notions of provisional, negative, and apparent convergence to 0 are introduced. It is then shown that Markov's principle is equivalent, in Bishop-style constructive mathematics, to the statement `every decreasing sequence of real numbers provisionally convergent to 0 actually converges to 0', and that this equivalence holds with `provisionally' replaced by `negatively'. Finally, apparent convergence and convergence are related by means of the anti-Specker principle.