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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)