
Re: [HM] A question as to Cantor's Diagonal Method.
Posted:
Aug 30, 2005 11:08 AM


Original Message From: William Tait Sent: Sunday, August 21, 2005 6:50 PM To: historiamatematica@chasque.apc.org Subject: Re: [HM] A question as to Cantor's Diagonal Method.
On Aug 18, 2005, at 3:54 AM, Alexander Zenkin wrote:
> Whether the following fact is known/described anywhere in mathematical > literature? > > Here X=[0,1], N={1,2,3, ...}, > RAA = Reductio ad Absurdum, > CDM = Cantor's Diagonal Method. > Z  a cardinality of a set Z for any Z. > > THEOREM 5. If X = N, then there is NOT a rule/algorithm > producing a 11correspondence between elements of the sets, X and N. > RAAPROOF. Assume that X = N, BUT there is a rule/algorithm > producing a 11correspondence between elements of X and N. Then the > rule/algorithm generates a list, > x1, x2, x3, . . . , (1) > containing ALL real numbers from X. Applying CDM to the list (1), > Cantor > generates a new real number, which doesn't belong to the list (1). > So, the > given list (1) contains NOT ALL real numbers from X. Contradiction. > Q.E.D. >
I believe that I noted this a while back in a communication to the FOM list, in pointing out that the 'diagonal method' has a positive content. (I expect that it has been pointed out many times before.) Note, however, that what is quoted above from Alexander's posting isn't quite Cantor's theorem (although it is surely an immediate consequence of it): Rather it is: given a function from a set M to 2^M, we can construct an element of 2^M not in the range of the function.
Regards to all,
Bill Tait
Dear Bill,
You are right  the Theorem 5 is not Cantor's theorem, since it doesn't state the uncountability of X. The Theorem 5 doesn't state the countability of X as well. The Theorem 5 is a conditional statement: IF X is countable, THEN no 11correspondence between elements of the sets, X and N, can be established. It is a quite unexpected statement, but if its RAAPROOF (above) has no fallacy as we, Cantor's diagonal method and me, hope :) , then the statement is true. And I would like to know whether this fact is described in mathematical literature.
Best wishes,
AZ

