 Then, in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind (see Ewald 1996, vol. 2), Cantor asked the question whether the infinite sets \(\mathbf{N}\) of the natural numbers and \(\mathbf{R}\) of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set \(A\) of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with \(\mathbf{N}\)). A few days later, Cantor was able to prove that the assumption that \(\mathbf{R}\) is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness mentioned above. Thus he had shown that there are more elements in \(\mathbf{R}\) than in \(\mathbf{N}\) or \(\mathbf{Q}\) or \(A\), in the precise sense that the cardinality of \(\mathbf{R}\) is strictly greater than that of \(\mathbf{N}\).