this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One may also infer that there are unions of three closed sets di#erent from every union of two closed sets. These observations are the tip of an iceberg. The intuitionistic Borel Hierarchy shows o# an exquisite fine structure

Brouwer's Continuity Principle distinguishes intuitionistic mathematics from other varieties of constructive mathematics, giving it its own avour. We discuss the plausibility of this assumption and show how it is used. We explain how one may understand its consequences even if one hesitates to accept it as an axiom. 1 Brouwer's Continuity Principle We let N be the set of all natural numbers. Its elements 0; 1; 2; : : : are produced one by one. N is a never nished project that is executed step-by-step. We let N be the set of all innite sequences of natural numbers. The acceptance of N as a totality has been a major step in the history of mathematical thinking, and led to the development of set theory. With Cantor's diagonal argument in mind, Brouwer probed the meaning of the words: \every possible innite sequence of natural numbers" and found a way to sensibly use them. An element of N is a function from N to N, = (0); (1); (2); : : : Every such element is produced st...

This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with the property that every constructive partial functional defined on {# : R(#)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(#) is equivalent in #) for some stable A(#, #) (which belongs to the classical analytical hierarchy). The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems.

et of all innite sequences of natural numbers. Every element of N is a function from N to N and in some sense an innite and incomplete object, a kind of project to be carried out or lled in in the future; its values (0); (1); (2); : : : are produced one by one, in a never ending sequence. We do not require that there exists a rule or a nitely described algorithm that denes the project, and makes that it could be left to itself, we ourselves take care that the project is continued and supply a next value every time, perhaps freely choosing one, perhaps following some more or less secret device of our own. We use m; n; p; : : : as variables over N and ; ; ; : : : as variables over N . 1 Wim Veldman 1.3. First Axiom of Countable Choice Let R be a subset of<