cantor

Cantor
1. A mathematician.
Cantor devised the diagonal proof of the uncountability of the real numbers:
Given a function, f, from the natural numbers to the real numbers, consider the real number r whose binary expansion is given as follows: for each natural number i, r's i-th digit is the complement of the i-th digit of f(i).
Thus, since r and f(i) differ in their i-th digits, r differs from any value taken by f. Therefore, f is not surjective (there are values of its result type which it cannot return).
Consequently, no function from the natural numbers to the reals is surjective. A further theorem dependent on the axiom of choice turns this result into the statement that the reals are uncountable.
This is just a special case of a diagonal proof that a function from a set to its power set cannot be surjective:
Let f be a function from a set S to its power set, P(S) and let U = x not in f(x) . Now, observe that any x in U is not in f(x), so U != f(x); and any x not in U is in f(x), so U != f(x): whence U is not in x in S . But U is in P(S). Therefore, no function from a set to its power-set can be surjective.
2. An object-oriented language with fine-grained concurrency.
[Athas, Caltech 1987. "Multicomputers: Message Passing Concurrent Computers", W. Athas et al, Computer 21(8):9-24 (Aug 1988)].
(1997-03-14)