Cantor
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Related to Cantor: canter, Cantor set
cantor
1. Judaism a man employed to lead synagogue services, esp to traditional modes and melodies
2. Christianity the leader of the singing in a church choir
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
Cantor
(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 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.
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(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.
Cantor
(language)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)].
[Athas, Caltech 1987. "Multicomputers: Message Passing Concurrent Computers", W. Athas et al, Computer 21(8):9-24 (Aug 1988)].
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.
Cantor
in the Catholic Church, a singer; in Protestant churches, a singing teacher, choir conductor, and organist, whose duties also often included the composition of music for the church (for example, J. S. Bach at St. Thomas in Leipzig). In a Jewish synagogue, the main singer, or hazan.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.