The mathematical idea of a Set (and a Set's intentional counterpart—a Relation) is at once the most important and the most enigmatic of all mathematical concepts. Not that there are not more difficult mathematical ideas, or even more fundamental. Certainly the Calculus is among the hardest of all mathematical subjects to understand. But Calculus, like almost all other areas of higher math, is based upon the theory of Sets. And of course the ideas of Sets and Relations thoroughly permeates the recently developed theory of mathematical Categories and Functors, which plays such a fundamental role in the organization of mathematics. Naturally, this fundamentality applies to the notion of Sets (not to mention Relations, as well.

But what exactly are Sets? Well, that is the real question. The founder of Set Theory, Georg Cantor, believed, at least for a while, that he knew the answer: a set, he said, is a Many thought of as One. But, unfortunately for Cantor, mathematicians soon discovered that this definition leads to contradictions, and so it had to be abandoned. Even worse, since that discovery no one, mathematician or not, has been able to give a definition of 'set' that is not mathematically pernicious in one way or another (if it is not contradictory, then perhaps it is too vague). As a result, nowadays the idea of a Set is typcally left undefined. This, however, is somewhat embarassing, since Sets play such a comprehensive role at the very foundation of much of Mathematics. All ideas about Numbers, for instance, can be explained in terms of Sets and the intensional counterparts, Relations. And of course Numbers, although certainly not the only objects in Mathematics, are among the most important. Number Theory, for instance, which ask such questions as "What are the Prime Numbers?", would obviously be impossible without Numbers themselves. And the theories of Fields and Rings, theories about orders Sets of Numbers, would be of little help to Science if the idea of a Set were shown to be a contradiction. Thus the Ontological question "Do Sets exists?" is among the more pertinent questions in philosophy.