Master Class 2006/2007 on Logic

Set-theoretical and constructive aspects of the continuum (Wim Veldman and Bas Spitters)

The modern study of the continuum may be said to have started with Cantor's discovery of the uncountability of the reals in 1873. This discovery, and the further work that followed, soon gave rise to a number of difficult questions, that, even today, have not found a final answer, like: The continuum hypothesis: is every uncountable subset of the reals equivalent with the whole set? Does there exist a well-ordering of the continuum? Is every subset of the reals Lebesgue-measurable?

We want to study these questions and the partial answers that have been found, some of them involving the axiom of choice. We also want to find out which sense these questions and answers could make from a constructive point of view. We shall consider Brouwer's view and his proposals to change the language and axioms of mathematics. We shall study what happens if one requires every real to be given by an algorithm. We will try to see what light is shed upon them by the so-called formal or pointless topology.