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Cantor

1. person, mathematics A mathematician.

Cantor devised the diagonal proof of the uncountability of the

Given a function, f, from the natural numbers to the realnumbers, 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 in S: 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 f(x) : x in S . But U is

in P(S). Therefore, no function from a set to its power-set

can be surjective.

2. language An object-oriented language with fine-grained

[Athas, Caltech 1987. "Multicomputers: Message Passing

Concurrent Computers", W. Athas et al, Computer 21(8):9-24

(Aug 1988)].