The Grzegorczyk hierarchy can be extended to transfinite ordinals. Georg Cantor would introduce the aleph symbol for cardinal numbers of transfinite sets. He called the cardinal numbers of these infinite sets transfinite cardinal numbers. Moreover, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations . Certain fields of mathematics define infinite and transfinite numbers. This is actually a definition by transfinite recursion. This is again a definition by transfinite recursion. Again, this is really a definition by transfinite recursion. The "transfinite" cardinal numbers describe the sizes of infinite sets. Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. In 1883, Cantor divided the infinite into the transfinite and the absolute. The uniqueness of the sequence satisfying these properties can be proved using transfinite induction. These give two different generalizations of addition of natural numbers to the transfinite. This more generalized version extends the aforementioned definition to encompass transfinite listings. All variants of induction are special cases of transfinite induction; see below.