Philip Welch

Infinity: scaling the heights

An extensive area of research in modern set theory concerns the relationship between so-called `large cardinal' numbers (as originally defined by Hausdorff) and the real continuum as studied by analysts. This "descriptive set theory" of the sets of reals definable in analysis depends on axioms that go beyond the usual ones of Zermelo-Fraenkel established in the 1920's. How are these two areas connected? Why should infinite cardinal numbers remote from the appearance of the real continuum affect at-a-distance the classes of sets that are, eg Lebesgue measurable? One key area is perhaps surprising: the existence of winning strategies for two person perfect information games, played in Baire space (going back to Banach, Ulam et al.) Such strategies determine both which sets are measurable, but also build inner models of set theory for large cardinals and strong axioms of infinity.