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In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of ...
Transfinite number, denotation of the size of an infinite collection of objects. Comparison of certain infinite collections suggests that they have different sizes even though they are all infinite. For example, the sets of integers, rational numbers, and real numbers are all infinite; but each is.
In his magisterial Grundlagen [1883] Cantor developed the transfinite numbers [Anzahlen] and the key concept of well-ordering. A well-ordering of a set is a linear ordering of it according to which every non-empty subset has a least element. No longer was the infinitary indexing of his trigonometric series investigations mere contrivance.
Transfinite numbers are one of Cantor's ordinal numbers , , , ..., , , ... all of which are "larger" than any whole number . As noted by Cantor in the 1870s, while it is possible to distinguish different levels of infinity, most of the details of this have not been widely used in typical mathematics.
Transfinite numbers are numbers that help us compare the size of infinite sets, such as the natural numbers and the real numbers.
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.
aleph-null (ℵ0), in mathematics, the cardinality of the infinite set of natural numbers {1, 2, 3, …}. The cardinality, or cardinal number, of a set is the number of elements of a set.
