transfinite number


Also found in: Dictionary, Wikipedia.

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
..... Click the link for more information.
 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.
..... Click the link for more information.
; 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.
References in periodicals archive ?
to study the relationships of transfinite numbers not only mathematically, but to document and investigate them in all places where they occur in nature.
46) This is apparent already from the initial segment of the transfinite numbers given as follows.
Nevertheless, Cantor's contention that in the formation of transfinite numbers via the first two generation principles "we seem to get lost in the unlimited" (49) is essentially correct.
54) Some inkling of this is provided by the absolutely unlimited growth of the sequence of transfinite numbers which Cantor regarded as an "appropriate symbol of the Absolute.
Moving beyond the generation of transfinite numbers to the formation of sets, we may conceive of the apeiron in Cantor as a general reference to the inexhaustibility of the operation "set of.
62) It may be argued that the transfinite numbers themselves are manifestations of peras since they represent "fixed boundaries" being "completed infmities.
Under this interpretation, set theory assumes the form of a dialectic which already manifests itself in the restricted setting of transfinite numbers in the opposition between the first two principles of generation and the principle of limitation.
There are questions to investigate about transfinite numbers, such as the truth of the continuum hypothesis, but not their real meaning--they too just are.
Is it possible that there are transfinite numbers between two alephs, especially between aleph-one and aleph-two?