26 ERRETT A. BISHOP natural metrics on the space D of domains (with distinguished points z ) and 0 the space M of mapping functions, and proving that the function A : D + M that associates to each domain its mapping function is a homeomorphism. He thus extends and constructivizes the classical Caratheodory results. Many of his estimates are similar to those developed by Warschawski in his studies of the ma,pping function. The problem of constructivizing the classical theory of uniformization is still open. (Even reasonable conjectures seem drnicult to come by. ) So is the problem of (constructively) constructing canonical maps for multiply con- nected domains, as far as I know. It is natural to define two sets to have the same cardinality if they are in one-one correspondence. The constructive theory of cardinality seems hopelessly involved, due to the constructive failure of the Cantor-Bernstein lemma, and for other reasons as well. Progress has been made however in constructivizing the theory of ordinal numbers. Brouwer [8] defines ordinals to be ordered sets that are built up from non-empty finite sets by finite and countable addition. F. Richman 1261 gives a more general definition. Simple in appearance, his definition constructivizes the property of induction in just the right way. An ordinal number (or well-ordered set) is a set S with a binary relation such that (1) if a b and b c, then a c (2) one and only one of the relations a b, b a, a = b holds for given elements a and b of S (3) let T be any subset of S with the property that every element b of S, such that a E T for each a in S with a b, belongs to T then T =S. Richman shows that each Brouwerian ordinal satisfies (1). (2), and (3). He gives examples of ordinals (in his sense) that are not Brouwerian. He shows that every subset of an ordinal is an ordinal (under the induced order). He uses his theory to constructivize the classical theorems of Zippin and Ulrn concerning existence and uniqueness of p-groups with prescribed invariants. The above examples might give the impression that the constructiviza- tion of classical mathematics always proceeds smoothly. I shall now give some other examples, to show that in fact it does not.
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