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