 At this point, one might be forgiven for thinking that there is no cardinal number greater than that of the natural numbers, just as there is no extended real number larger than \(+\infty\). However, Cantor’s second striking result is that the cardinality of the positive real numbers is in fact greater than the cardinality of the positive integers, and his third striking result is that for every set, the set of all its subsets—its power set—has an even greater cardinality. Although many different infinite sets of different order types can all be put into one-to-one correspondence with each other, there are some infinite sets that cannot. Sets whose cardinality is equal to that of the natural numbers (like the integers and the rationals) are said to be countably infinite or denumerable, while infinite sets that are not countable (like the reals, and the power set of the naturals) are said to be uncountable. See proofs of the results in the