[FOM] RE: [HM] Cantor's diagonal proof
alexzen at com2com.ru
Sat Mar 13 23:33:32 EST 2004
On Saturday 28 Feb 2004 7:41 pm, Alexander Zenkin wrote:
>> A student of mine offered the following idea.
>> A key point of Cantor's diagonal proof of the uncountability
>> of continuum is an explicit usage of the method of counter-example.
>> Indeed, according to the known assumption of Cantor's proof, there
>> is a list, x1,x2,x3, . . . (1), containing ALL reals from X=[0,1].
>> Then the application of Cantor's diagonal method to the list (1)
>> generates a NEW real, say x*, from X which differs from every real
>> of the list (1). From the point of view of classical logic and
>> mathematics, this new real x* is a COUNTER-EXAMPLE disproving a
>> COMMON statement that "the given list (1) contains ALL reals from X".
>> Whether Cantor (or anybody else) was considering ever and
>> anywhere such aspect of the diagonal proof?
>> Thanks in advance,
>> Alexander Zenkin
On Monday 01 Mar 2004, Martin Davis wrote:
>Given any one-one correspondence between the
>natural numbers and a SPECIFIED set of real
>the diagonal method provides a counter-example
>in precisely Zenkin's sense.
I believe that Martin Davis is surely right: Cantor’s NEW real x* is just a
COUNTER-EXAMPLE disproving a COMMON statement that "the given list (1)
contains ALL reals from X".
But my question was about “whether Cantor (or anybody else) was considering
ever and anywhere such aspect of the diagonal proof?”
Since nobody of [HM]-members answered the question positively, I believe
that my student made a real meta-mathematical discovery the essense of which
STUDENT’S DISCOVERY. The Cantor famous Diagonal Method (in its any
meta-mathematical realizations) is a special case of the Counter-Example
Method where a counter-example itself is not searched for in a set of all
possible realizations of a given common statement, but is formally deduced
from that common statement, which this counter-example must disprove.
Remark that in the traditional counter-example method a counter-example is
just searched for in a set of all possible realizations of a given common
statement. E.g., by means of the Monte-Carlo method.
Any views are welcome.
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