The Utility of the Uncountable

In my lecture at the 2011 Congress on Logic, Methodology, and the Philosophy of Science in Nancy, France, I spoke on an additional axiom of set theory — the Proper Forcing Axiom — which has proved very successful in settling combinatorial problems concerning uncountable sets. Since I have already written a exposition on this subject [43], I have decided to address a broader question in this article: why study uncountability? In some circles within logic, there has been an ongoing campaign to stress the importance of countability in mathematics — and to marginalize the uncountable. While much of mathematics does concern objects which can be codified as hereditarily countable sets, this often does not reflect how mathematics is discovered or developed. More significantly, there are technical difficulties which can arise in mathematics — often quite unexpectedly — which are fundamentally uncountable in their character. The purpose of this article is survey some instances where uncountability has been useful in the discovery process, essential to the solution of a problem, or at least has offered a fruitful perspective. We will also will examine settings in which restricting attention to countable objects artificially limits the perspective and gives an incomplete picture of the mathematical phenomenon under consideration. In this article, we will take countable mathematics to mean the study of that which can be encoded in the hereditarily countable sets — the domain of discourse of second order arithmetic. For instance a complete metric space can be encoded as the completion of a countable metric space. Even Borel or suitably definable subsets of such a space have a countable description and as such lie within the scope of “countable mathematics.” Nonseparable spaces or nonmeasurable subsets of R are typical examples of objects which are essentially uncountable in their nature.