Set theory forms the foundation of the entire edifice of mathematics. It is far from elementary: how can a set be defined rigorously? A set is, of course, made up of a "collection of objects." For each set, we must then determine precisely which objects belong to it and which do not. How can this be done? The simplest way to define a set is extensionally. We simply list the elements of the set as a collection of objects. Thus, the possible outcomes of rolling a die form a set E consisting of the natural numbers 1, 2, 3, 4, 5, and 6. We write it as E = {1, 2, 3, 4, 5, 6}.
Another approach is to define the elements of the set intensionally. We need only characterize all the elements we wish to include in the set by specifying one or more properties they share. This approach can sometimes cause problems, even when the properties in question seem clear.
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From the empty set to the natural numbers
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The easiest set to define extensionally is the set containing no elements: the empty set, denoted by the Scandinavian letter . The origin of this notation is well known: a symbol was needed that resembled 0 without actually being 0. The French mathematician André Weil, who knew Norwegian, introduced this notation in 1937. Here, then, is the empty set:
