Abstract

The thesis examines two dimensions of constructivity that manifest themselves within foundational systems for Bishop constructive mathematics: intuitionistic logic and predicativity. The latter, in particular, is the main focus of the thesis. The use of intuitionistic logic affects the notion of proof : constructive proofs may be seen as very general algorithms. Predicativity relates instead to the notion of set: predicative sets are viewed as if they were constructed from within and step by step. The first part of the thesis clarifes the algorithmic nature of intuitionistic proofs, and explores the consequences of developing mathematics according to a constructive notion of proof. It also emphasizes intra-mathematical and pragmatic reasons for doing mathematics constructively. The second part of the thesis discusses predicativity. Predicativity expresses a kind of constructivity that has been appealed to both in the classical and in the constructive tradition. The thesis therefore addresses both classical and constructive variants of predicativity. It examines the origins of predicativity, its motives and some of the fundamental logical advances that were induced by the philosophical re ection on predicativity. It also investigates the relation between a number of distinct proposals for predicativity that appeared in the literature: strict predicativity, predicativity given the natural numbers and constructive predicativity. It advances a predicative concept of set as unifying theme that runs across both the classical and the constructive tradition, and identifies it as a forefather of a computational notion of set that is to be found in constructive type theories. Finally, it turns to the question of which portions of scientifically applicable mathematics can be carried out predicatively, invoking recent technical work in mathematical logic.