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Re: Cantor Paradox
Posted:
Mar 11, 2004 12:21 PM


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 >nth digit of the nth 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



