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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.