At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intuitionistic reader. 1.

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.

Department of Philosophy-Utrecht University Ramsey's theorem and the pigeonhole principle in intuitio flistjc mathematics