Constructive mathematics without choice

What becomes of constructive constructive mathematics without the axiom of (countable) choice? Using illustrations from a variety of areas, it is argued that it becomes better. Despite the apparent unanimity among schools of constructive mathematics with respect to the acceptance of the countable axiom of choice, I believe it to be one of the central problems, or mysteries, of constructive mathematics. The axiom may be written as ∀m∃nP (m,n) =⇒ ∃α ∀mP (m,αn) (CC) where P is a binary predicate, m and n are in N, and α ∈ N. Brouwer’s choice sequences constitute one approach to this problem. Such sequences may be thought of as finite sequences that have the potential of going on indefinitely. That is, they embody the hypothesis of CC (or even the axiom of dependent choices): the ability to go on.. We could say that Brouwer solved the problem of CC by defining the conclusion to be the hypothesis. Other justifications of CC, including some by intuitionists, consider how we could know that the hypothesis was true, and conclude that we would have to possess an infinite sequence α as in the conclusion. We could say that this solves the problem of CC by defining the hypothesis to be the conclusion. I would like to suggest that the way to approach the problem is in accordance with Bishop’s fourth principle of constructivism, that meaningful 01981 Mathematics Subject Classification. Primary 03F65, 03E25