Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.

In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor’s continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The smallest infinite number, ℵ0, ‘aleph-nought, ’ gives the number of positive whole numbers. A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps. Cantor’s revolutionary discovery was that the points on a line cannot be so listed, and so the number of points on a line is a strictly higher infinite number (c, ‘the cardinality of the continuum’) than ℵ0. Hilbert’s First Problem asks whether any infinite subset of the real line is of one of these two cardinalities. The axiom that this is indeed the case is known as the Continuum Hypothesis (CH). This problem had unexpected connections with Hilbert’s Second Problem (and even with the Tenth, see the article by M. Davis and the comments on