The Borel Hierarchy Theorem from Brouwer's intuitionistic perspective

In intuitionistic analysis, Brouwer's Continuity Principle implies, together with an Axiom of Countable Choice, that the positively Borel sets form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level. ?0. Introduction. 0.1. ?. Borel, H. Lebesgue, R. Baire, N. Lusin, A. Souslin and others, the founding fathers of descriptive set theory, who initiated the study of Borel sets and, somewhat later, discovered analytic and projective sets, pursued their subject not only out of a mathematician's curiosity but also from a sense of bewilderment characteristic of the philosopher. Frowning at some of the notions and arguments in Cantorian set theory, they wanted to develop an, in their own words, realist point of view. Others, however, have called them semi-intuitionists, see [2]. These semi-intuitionists doubted, for instance, the existence of the choice set lying at the basis of Zermelo's proof of the Well-Ordering Theorem, as no one is able to give a description of such a set, and also the existence of Cantor's second number class, that is, the first uncountable ordinal Hi, as no one can imagine a point of time where the construction of its members would be finished, see [6] and [32], One may be surprised that they nevertheless were prepared to accept the continuum, that is, the set R of the real numbers, as somehow given by geometric intuition. They did not see, however, how to attach a sense to the expression: "all subsets of the set M" Received May 15, 2002. ? 2008. Association for Symbolic Logic 0022-4812/08/7301-0001/S7.40