Mathematics – Logic
Scientific paper
2011-04-15
Mathematics
Logic
82 pages
Scientific paper
We study projective subsets of Baire space from Brouwer's intuitionistic point of view, using his Thesis on Bars and his continuity axioms. We first study analytic sets; these are the projections of the closed subsets of Baire space. We consider a number of examples and discover a fine structure in the class of the analytic sets that fail to be positively Borel. A subset of Baire space is strictly analytic if it coincides with the range of a continuous function from Baire space to itself. We prove separation and boundedness theorems for strictly analytic sets. Co-analytic sets are the co-projections of the open subsets of Baire space. We show different ways to prove that some co-analytic sets are not analytic and that some analytic sets are not co-analytic. We consider the set of the codes of the closed and located subsets of Baire space that are almost-countable as an example of a set that is a projection of a co-analytic set. We bring to light the collapse of the projective hierarchy: every (positively) projective set coincides with the projection of a co-analytic set.
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