"Unless it were a transfinite number
falling in love with a finite one--I suppose such things do happen, even in mathematics."
The group of real numbers represents a higher infinity, a transfinite number
. Furthermore, all real numbers correspond to the points in a line, so that points are also nondenumerable.
Even so, he spent much of his later life addressing ancient and scholastic objections to his own transfinite number
(47) Here [omega] denotes the smallest transfinite number
which Cantor conceived of as the least upper bound of the progression of integers, as expressed in the second principle of generation.
If aleph-n is a small transfinite number
, then so is aleph-(n + 1).
Borges proposes a clear explanation of Russell's lesson about the transfinite numbers
in "El tiempo" (Borges oral).
Through the rest of chapter 3, Maddalena further explicates thought's continuity using tools from mathematics, following the lead of Peirce, who famously appreciated Cantor's controversial discovery of transfinite numbers
, attempting to engage the brilliant schoolteacher in argument about the greatest infinity.
This applies both to cosmological phenomena such as singularities, which are often considered bodies of infinite mass and density, and to "inventions" such as Cantor's system of transfinite numbers
(I understood Cantor's theory of transfinite numbers
and was sure that Korzybski didn't.) Korzybski identified sanity with the mode of thinking that evolved in the natural sciences and in mathematics, and so did Bertrand Russell (if "sanity" is taken as a synonym of "rationality-cum-morality"), and so did Hayakawa (if pragmatism is assumed to be the psychological substrate of sanity).
Acting out of obedience to carry out his understanding of God's will, Cantor developed a theory of transfinite numbers
. It was vigorously opposed by well-known mathematicians such as Leopold Kronecker, who, like Brouwer, was an Intuitionist (see Section 1.1, Logic).
"Such a study would embrace the mutual relationships of the continuous and discontinuous, the real meaning of surds [irrational numbers] and transfinite numbers
, the infinitesimal, non-Euclidean space, and the validity of mathematical transcripts of physical reality such as quantum mechanics and the theory of relativity" (12-13).