# 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

_{0}

*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, …

*n*

^{2}, … }; 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,

_{0}

_{0}

_{0}

_{0}

_{0}

_{0}

_{0}

_{0}

*n*×

_{0}

_{0}

_{0}

^{ n }=

_{0}

*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

_{0}

*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.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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