The idea of counting is one of the first mathematical ideas ever considered, and spawned many fundamental areas of mathematics such as number theory or combinatorics. The idea of counting is essentially that of putting some collection of objects in bijection with some finite set. However, we might also want to consider infinite sets, and in that case the idea of size of a set leads to different concepts for infinite sets: ordinals and cardinals. Ordinals and cardinals of finite sets are essentially the same thing, but differences appear when considering infinite sets. Their importance is then crucial to understanding how large infinite sets are, and allow us to formulate precisely how much larger some infinite sets are than others; for example we know that the set of real numbers cannot be put in bijection with the set of rational numbers; indeed, the first is uncountable whereas the second is countably infinite. But there also exists different sizes of uncountable sets, and this idea is fundamental to the concept of infinite cardinals.

There are many interesting problems that can be formulated in terms of ordinals and cardinals, probably the most famous one being the Continuum hypothesis, which has been proven to be independent for the currently accepted set of axioms for set theory (Zermelo-Fraenkel set theory).

Join us at 7:30 in MS.04 while Cosmin Davidescu explains all this and even some applications to some seemingly unrelated problems, for example in complex analysis.