<Li> Any set X that has the same cardinality as the set of the natural numbers, or X = N = א 0 (\ displaystyle \ aleph _ (0)), is said to be a countably infinite set . </Li> <Li> Any set X with cardinality greater than that of the natural numbers, or X> N, for example R = c (\ displaystyle (\ mathfrak (c)))> N, is said to be uncountable . </Li> <P> Our intuition gained from finite sets breaks down when dealing with infinite sets . In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view that the whole cannot be the same size as the part . One example of this is Hilbert's paradox of the Grand Hotel . Indeed, Dedekind defined an infinite set as one that can be placed into a one - to - one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite . Cantor introduced the cardinal numbers, and showed that (according to his bijection - based definition of size) some infinite sets are greater than others . The smallest infinite cardinality is that of the natural numbers (א 0 (\ displaystyle \ aleph _ (0))). </P> <P> One of Cantor's most important results was that the cardinality of the continuum (c (\ displaystyle (\ mathfrak (c)))) is greater than that of the natural numbers (א 0 (\ displaystyle \ aleph _ (0))); that is, there are more real numbers R than whole numbers N. Namely, Cantor showed that c = 2 א 0 = ב 1 (\ displaystyle (\ mathfrak (c)) = 2 ^ (\ aleph _ (0)) = \ beth _ (1)) (see Beth one) satisfies: </P>

Which of the following sets is the super set for the other three sets