The Continuum Problem is whether there is a set of reals whose cardinality is strictly between the cardinality of the integers and the reals.
This was the first problem on Hilbert’s famous list and it turned out to be undecidable by the usual axiom systems for Set Theory. The results of Gödel and Cohen tell us that the axioms give very little information about the relative size of the set of integers and the set of reals. Gödel’s conjecture that strong axioms of infinity will settle the problem turned out to be false. Is this the end of the story? In this talk we shall survey some of current approaches of trying to give a meaningful answer tothe problem, in spite of its independence. We shall concentrate on the theory of universally Baire set of reals , which tries to formalize the concept of a set of reals being ”regular”, ”non-pathological” etc.