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Topic: Re: Cantor Paradox
Replies: 5   Last Post: Mar 14, 2004 1:43 PM

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Danny Purvis

Posts: 176
Registered: 12/6/04
Re: Cantor Paradox
Posted: Mar 11, 2004 12:21 PM
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On 11 Mar 2004, Rupert wrote:

>Nathan's list is the list of all real numbers definable by a finite
>string. He must specify definable in what language, say L.
>
>Then the issue is whether the diagonal number for Nathan's list can
>be defined in L. Its definition contains such expressions as "the
>n-th digit of the n-th number defined in L" and it is not clear that
>these can be defined in L.
>
>Since a contradiction results from the supposition that the diagonal
>number can be defined in L, Nathan's paradox actually works as a
>reductio ad absurdum to show that it cannot be defined in L.


Well, Nathan evidently thinks it IS clear that the diagonal can be
defined in L. Let L be English. He evidently thinks he has defined
in English exactly how we go about constructing the diagonal.

The distinction between paradox and reductio ad absurdum can be
controversial. A mathematical argument reaching an absurd conclusion
calls into question the underlying assumptions of the argument, but
there can be disagreements about which underlying assumption is
faulty. If there is only one possible candidate, the argument is a
reductio ad absurdum. But if it not clear that there is only one
possible candidate then, until we figure things out, we should use
the word "paradox".

Nathan evidently does not believe that an ability to describe his
diagonal in L is obviously the sole possible candidate for
faultiness, and so he evidently finds it appropriate to call into
question other propositions upon which his argument is based.

Perhaps Nathan thinks that diagonalization in general is suspect.
Perhaps he thinks there is some problem in the concept of infinity or
of denumerability. Perhaps he thinks there is some much more deeply
rooted proposition whose faultiness he has uncovered. Perhaps he
even thinks there is paradox at the heart of the mathematical
enterprise.


Danny Purvis




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