transfinite number

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 ℵ ₀ (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, … 2 n,  … }, 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, ℵ ₀ . 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 ℵ ₀ . Since the union of two countably infinite sets is a countably infinite set, ℵ ₀  + ℵ ₀  = ℵ ₀ ; moreover, ℵ ₀  × ℵ ₀  = ℵ ₀ , so that in general, n  × ℵ ₀  = ℵ ₀ and ℵ ₀ⁿ  = ℵ ₀ , 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 ℵ ₀ ; 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 2 ^c , 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.