Intuitionism and effective descriptive set theory

This is an elementary, mostly expository paper, broadly aimed at students of logic and set theory who also know the basic facts about recursive functions but need not know a lot about either intuitionism or descriptive set theory. The only (possibly) new result is Theorem 6.1, which justifies simple definitions and proofs by induction in Kleene's Basic System of intuitionistic analysis, and is then used inTheorem 7.1, Theorem 7.2 to give in the same system a rigorous definition of the Borel sets and prove that they are analytic; the formulation and proof of this last result is one example where methods from effective descriptive set theory are used in an essential way.