Cantor
1. <person, mathematics> 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 ith digit is the complement of the ith digit of f(i).
Thus, since r and f(i) differ in their ith 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 powerset can be surjective.
2. <language> An objectoriented language with finegrained concurrency.
[Athas, Caltech 1987. "Multicomputers: Message Passing Concurrent Computers", W.
Athas et al, Computer 21(8):924 (Aug 1988)].
(19970314)
Nearby terms:
canonicity « C (ANSI) « can't happen « Cantor
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