Unavoidable sequences in constructive analysis

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 2 . These are serious obstructions to interpreting the constructive content of classical analysis, just as the collapse of the arithmetical hierarchy at Σ 3 in HA + MP0 + ECT0 (cf. [6]) limits the scope and effectiveness of recursive analysis. Bishop’s constructive mathematics, now undergoing (partial) formalization, is consistent with intuitionistic analysis and also with recursive analysis so must have similar defects. It seems natural to ask whether e.g. intuitionistic analysis could incorporate more of classical mathematics without seriously compromising its constructive content. Brouwer and Bishop agreed that constructive mathematics was an intellectual work in progress. Bishop and Markov agreed on the primary importance of computational content. All three recognized the constructive significance of continuity. Their insights can be interpreted as prescribing admissible rules, rather than restrictive axiom schemas, for constructive formal systems compatible with larger parts of classical mathematics. A theory based on intuitionistic logic may adhere to a constructive closure rule without proving the corresponding implication. For example, the recursive choice rule known as Church’s Rule for arithmetic CR0: “If ∀x∃yA(x, y) is provable where A(x, y) is arithmetical and contains only x, y free, then ∃e∀x∃y∃z[T(e, x, y) & U(y) = z & A(x, z)] is also provable.” holds for intuitionistic arithmetic HA, while the arithmetical form CT0 of Church’s Thesis is unprovable. Similarly, HA satisfies Markov’s Rule for arithmetic MR0: “If ∀x(A(x) ∨ ¬A(x)) & ¬¬∃xA(x) is provable then also ∃xA(x) is provable.” but does not prove the corresponding implication MP0. One type up, a constructive theory of numbers and number-theoretic sequences (“constructive analysis”) based on intuitionistic logic generally satisfies Brouwer’s Rule of continuous choice, some form of Markov’s Rule, and the Church-Kleene Rule asserting that only recursive sequences can be proved to exist; precise definitions are in the next section.