1.4.11: Summary
The size of a set \(A\) can be measured by a natural number if the set is finite, and sizes can be compared by comparing these numbers. If sets are infinite, things are more complicated. The first level of infinity is that of countably infinite sets. A set \(A\) is countable if its elements can be arranged in an enumeration , a one-way infinite list, i.e., when there is a surjective function \(f\colon \PosInt \to A\) . It is countably infinite if it is countable but not finite. Cantor’s zig-zag method shows that the sets of pairs of elements of countably infinite sets is also countable; and this can be used to show that even the set of rational numbers \(\Rat\) is countable.
There are, however, infinite sets that are not countable: these sets are called uncountable . There are two ways of showing that a set is uncountable: directly, using a diagonal argument , or by reduction . To give a diagonal argument, we assume that the set \(A\) in question is countable, and use a hypothetical enumeration to define an element of \(A\) which, by the very way we define it, is guaranteed to be different from every element in the enumeration. So the enumeration can’t be an enumeration of all of \(A\) after all, and we’ve shown that no enumeration of \(A\) can exist. A reduction shows that \(A\) is uncountable by associating every element of \(A\) with an element of some known uncountable set \(B\) in a surjective way. If this is possible, then a hypothetical enumeration of \(A\) would yield an enumeration of \(B\) . Since \(B\) is uncountable, no enumeration of \(A\) can exist.
In general, infinite sets can be compared sizewise: \(A\) and \(B\) are the same size, or equinumerous , if there is a bijection between them. We can also define that \(A\) is no larger than \(B\) ( \(\cardle{A}{B}\) ) if there is an injective function from \(A\) to \(B\) . By the Schröder-Bernstein Theorem, this in fact provides a sizewise order of infinite sets. Finally, Cantor’s theorem says that for any \(A\) , \(\cardless{A}{\Pow{A}}\) . This is a generalization of our result that \(\Pow{\PosInt}\) is uncountable, and shows that there are not just two, but infinitely many levels of infinity.