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cantor

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## cantor

```  3  definitions  found

From  Webster's  Revised  Unabridged  Dictionary  (1913)  [web1913]:

Cantor  \Can"tor\,  n.  [L.,  a  singer,  fr  caner  to  sing.]
A  singer;  esp.  the  leader  of  a  church  choir;  a  precentor.

The  cantor  of  the  church  intones  the  Te  Deum.  --Milman.

From  WordNet  r  1.6  [wn]:

cantor
n  1:  the  musical  director  of  a  choir  [syn:  {choirmaster},  {precentor}]
2:  the  official  of  a  synagogue  who  conducts  the  liturgical  part
of  the  service  and  sings  or  chants  the  prayers  intended  to
be  performed  as  solos  [syn:  {hazan}]

From  The  Free  On-line  Dictionary  of  Computing  (13  Mar  01)  [foldoc]:

Cantor

1.    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.    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)

```

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