1 October 2003, General Mathematics Colloquium, Benedikt Löwe(UvA, ILLC)

In the everyday experience of the average mathematician, there are two infinite cardinalities: countable (like the natural numbers) and uncountable (like the real numbers). It is just an empirical fact that whenever you pick an infinite set of real numbers, it's either countable or there is a rather easily definable bijection with the entire set of real numbers. And yet, set theorists know an infinitude of cardinalities, and logicians claim that so-called large cardinals have an influence on the foundations of mathematics, and even on concrete mathematical questions about concrete mathematical objects (e.g., the real numbers). What are large cardinals? What is the correlation between them and the theory of the real numbers? And why do we rarely (if ever) see those sets of real numbers that are influenced by the existence or nonexistence of these huge objects?