uncountable set

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uncountable set

[¦ən′kau̇nt·ə·bəl ′set]
(mathematics)
An infinite set which cannot be put in one-to-one correspondence with the set of integers; for example, the set of real numbers.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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In 1873, the German mathematician Georg Cantor published a paper in the Crelle Journal which proved that the set R of the continuum of real numbers is non-denumerable; that is, there is no one-to-one correspondence from the set N to the set R.
109], that "Cantor's set theory is so copious as to admit absolutely non-denumerable sets while axiomatic set theory [in particular, ZFC] is so limited [Skolem's paradox] that every non-denumerable set becomes denumerable in a higher system or in an absolute sense".
It is clear that the k-set [P.sub.k]([omega]) is absolutely non-denumerable. THEOREM says that any k-set y of [S.sub.k] is such that y [subset or equal to] [omega] (i.
The difference here is not trivial: to perceive the infinite as a limit is of ancient provenance, whereas to perceive the infinite specifically as denumerable or non-denumerable has a lot to do with the rise of a modern, explicitly infinite, mathematics.
The position is that there are infinite, non-denumerable strange attractors embedded in the implicate Order of mass-energy.
The first major idea to emerge that later could be used to describe evolutionary processes is the mathematical theory of probability for describing processes that have an infinite, non-denumerable "number" of possible outcomes.
(18) An important feature of the ordinal [[omega].sub.1] is that, because it cannot be put into a one-to-one correspondence with the denumerable natural numbers, it is non-denumerable or uncountable.

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