Cantor
1. Cs> 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 in S: x not in f. 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. 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) In addition suitable Contents:
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