transfinite number, cardinal or ordinal number designating the magnitude (power) or order of an infinite set; the theory of transfinite numbers was introduced by Georg Cantor in 1874. The cardinal number of the finite set of integers {1, 2, 3, … n} is n, and the cardinal number of any other set of objects that can be put in a one-to-one correspondence with this set is also n; e.g., the cardinal number 5 may be assigned to each of the sets {1, 2, 3, 4, 5}, {2, 4, 6, 8, 10}, {3, 4, 5, 1, 2}, and {a, b, c, d, e}, since each of these sets may be put in a one-to-one correspondence with any of the others. Similarly, the transfinite cardinal number ℵ0 (aleph-null) is assigned to the countably infinite set of all positive integers {1, 2, 3, … n, … }. This set can be put in a one-to-one correspondence with many other infinite sets, e.g., the set of all negative integers {−1, −2, −3, … −n, … }, the set of all even positive integers {2, 4, 6, … 2n, … }, and the set of all squares of positive integers {1, 4, 9, … n2, … }; thus, in contrast to finite sets, two infinite sets, one of which is a subset of the other, can have the same transfinite cardinal number, in this case, ℵ0. It can be proved that all countably infinite sets, among which are the set of all rational numbers and the set of all algebraic numbers, have the cardinal number ℵ0. Since the union of two countably infinite sets is a countably infinite set, ℵ0 + ℵ0 = ℵ0; moreover, ℵ0 × ℵ0 = ℵ0, so that in general, n × ℵ0 = ℵ0 and ℵ0n = ℵ0, where n is any finite number. It can also be shown, however, that the set of all real numbers, designated by c (for “continuum”), is greater than ℵ0; the set of all points on a line and the set of all points on any segment of a line are also designated by the transfinite cardinal number c. An even larger transfinite number is 2c, which designates the set of all subsets of the real numbers, i.e., the set of all {0,1}-valued functions whose domain is the real numbers. Transfinite ordinal numbers are also defined for certain ordered sets, two such being equivalent if there is a one-to-one correspondence between the sets, which preserves the ordering. The transfinite ordinal number of the positive integers is designated by ω.
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