Definition
To define the beth numbers, start by letting
be the cardinality of any countably infinite set; for concreteness, take the set of natural numbers to be a typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define
which is the cardinality of the power set of A if is the cardinality of A.
Given this definition,
are respectively the cardinalities of
so that the second beth number is equal to, the cardinality of the continuum, and the third beth number is the cardinality of the power set of the continuum.
Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ:
One can also show that the von Neumann universes have cardinality .
Read more about this topic: Beth Number
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)