# transfinite number

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## 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, …

*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, 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

_{0}

^{n}= 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.

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## transfinite number

[tranz′fī‚nīt ′nəm·bər] (mathematics)

Any ordinal or cardinal number equal to or exceeding aleph null.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

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