transfinite number

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Related to Transfinite ordinal: Transfinite cardinal

transfinite number,

cardinal or ordinal numbernumber,
entity describing the magnitude or position of a mathematical object or extensions of these concepts. The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their
designating the magnitude (power) or order of an infinite setset,
in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g.
; 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.

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.
References in periodicals archive ?
The peras displays itself most directly in Grundlagen by way of the Hemmungs- oder Beschrankungsprinzip (principle of limitation) "through which certain limits are successively imposed upon the thoroughly endless generation process" (61) of transfinite ordinals allowing thereby quantitative comparisons.
For example, Gentzen's proof of the consistency of PA uses induction as far as the transfinite ordinal [[Epsilon].sub.0] (which is the limit of the ordinals [Omega], [[Omega].sup.[Omega]], [[Omega].sup.[[Omega].sup.[Omega]]], etc.).
This seems a total muddle, confusing the fact that if [[Omega].sup.*] is the first transfinite ordinal in reverse order, then [[Omega].sup.*] + 1 is a new collection having the same order type and cardinality as [[Omega].sup.*].

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