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The set of real algebraic numbers is countably infinite (assign to each formula its Gödel number.) So the cardinality of the real algebraic numbers is . Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is .
He also proved that is equal to , the cardinality of the power set of the natural numbers. The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers, , or alternatively, that =. [1]
In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set of all real numbers. The latter cardinal is denoted or .
The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, S, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S.
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.
Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real numbers. The latter cardinality is equal to the cardinality of the set of all subsets of a countably infinite set.
so that the second beth number is equal to , the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number is the cardinality of the power set of the continuum.
There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.