Georg Cantor's Theory of Transfinite Numbers


"Transfinite" is descended from Latin words meaning, roughly, "beyond limits." The HarperCollins Dictionary of Mathematics describes "transfinite number" as follows:

"A cardinal or ordinal number used in the comparison of infinite sets, the smallest of which are respectively the cardinal (Aleph -null) and the ordinal (omega). The set of rationals and the set of reals have different transfinite cardinality."

Cantor demonstrated that infinite numbers exist, and that some are, contrary to intuitive expectations, "bigger" than other infinite numbers. He showed that infinite subsets of the natural numbers (such as the set of perfect squares) can be put into one- to- one correspondence with the set of natural numbers; therefore, the number of members of such subsets must be the same as the number of elements in the set of natural numbers. Also, with Diagonal theorem, he showed that the set of rational numbers (i.e., fractions) can be put into one-to-one correspondence with the natural numbers, and therefore has the same cardinal number as the set of natural numbers. The transfinite cardinal of these sets is (sometimes called E), the "smallest" transfinite number.