Kleene’s formalization FIM of intuitionistic analysis ([3] and [2]) includes bar induction, countable and continuous choice, but is consistent with the statement that there are no non-recursive functions ([5]). Veldman ([12]) showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions

Vesley [1965], as extended by Kleene [1969]) includes bar induction, countable and continuous choice, but cannot prove that the constructive arithmetical hierarchy is proper. Veldman showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions to interpreting the constructive content of classical analysis, just as the collapse of the arithmetical hierarchy at Σ 0 3 in HA + MP0 + ECT0 limits the scope and effectiveness of recursive analysis. Question: Can we do better by working within classical extensions of nonclassical theories, or within classically correct theories obeying e.g. Church’s Rule or Brouwer’s Rule? We work in a two-sorted language L with variables over numbers and one-place number-theoretic functions (choice sequences). Our base theory M – the minimal theory used by Kleene [1969] to formalize the theory of recursive partial functionals, function

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