cantor

cantor [Lat.,=singer], a singer or chanter, especially one who performs the solo chants of a church service. The office of cantor, at first an honorary one, originated in the Jewish synagogues, in which from early times it was the custom to appoint a lay member to represent the congregation in prayer. The notation of the chants was forbidden. In the 6th cent. poetic prayer forms were developed, and with them more complicated modes, or music, thus necessitating professional cantors. In the early Christian church, cantors known as precentors had charge of the musical part of the service. In modern Roman Catholic and Anglican services cantors sing the opening words of hymns and psalms.

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cantor

In Judaism and Christianity, an official in charge of music or chants. In Judaism the hazzan (cantor) leads liturgical prayer and chanting. In medieval Christianity the cantor had charge of a cathedral's music—specifically, of supervising the choir's singing. The term also designated the head of a college of church music.

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1.	(person, mathematics)	Cantor - 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.
2.	(language)	Cantor - 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)].