Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis

Abstract

The following is a self-contained proof theoretic treatment of two of the principal axiom schemata of current intuitionistic analysis: the axiom of bar induction (Brouwer's bar theorem) and the axiom of continuity. The results are formulated in terms of formal derivability in elementary intuitionistic analysis H(§ 1), so the positive (i.e., derivability) results also apply to elementary classical analysis Z1 (Appendix 1). Both schemata contain the combination of quantifiers νfΛn, where f, g, … are intended to range over free choice sequences of suitable kinds of objects x, y, …; for example, natural numbers or sequences of natural numbers, and n, m, p, r, … over natural numbers (non-negative integers).