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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null , the smallest infinite cardinal In mathematics , a cardinal number , or cardinal for short, is what is commonly called the number of elements of a set .
The cardinality of a set may also be called its size, ... This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem.
Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. Kanamori, Akihiro; Magidor, M. (1978).
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ω α).
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers, sometimes called the continuum.It is an infinite cardinal number and is denoted by (lowercase Fraktur "c") or | | [1]
Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set [9] (e.g., "the third man from the left" or "the twenty-seventh day of January").
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Notably, ℵ ω is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers 2 ℵ 0: For any natural number n ≥ 1, we can consistently assume that 2 ℵ 0 = ℵ n, and moreover it is possible to assume that 2 ℵ 0 is as least as large ...