Two simple sets that are not positively Borel

The author proved in his Ph.D. Thesis [W. Veldman, Investigations in intuitionistic hierarchy theory, Ph.D. Thesis, Katholieke Universiteit Nijmegen, 1981] that, in intuitionistic analysis, the positively Borel subsets of Baire space N form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level. It follows from this result that there are natural examples of analytic and also of co-analytic sets that are not positively Borel. It turns out, however, that, in intuitionistic analysis, one may give surprisingly different and, in some sense, much more simple examples of analytic and co-analytic sets that fail to be positively Borel. In the paper, two such examples are given. In proving them correct, one obtains new proofs of the Borel Hierarchy Theorem. Brouwer’s Continuity Principle plays a crucial role in arguments.