On 2010-06-19, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: > So using this logic and applying it to Cantor's original proof, > there may also be lists of all Reals, but there is no finite > algorithm which can generate the list?
No, because the antidiagonal is always real and not on the list. It need not be computable.
> How do you know that Cantor's proof that that there can be no list > of all Reals is simply because there is no finite algorithm to > produce the list, and not because they are uncountable?
For the simple reason that Cantor's proof makes no assumption of any finite algorithm and still concludes that the list does not contain all reals. Hence it works even for uncomputable lists.
> Cantor's proof requires a purported list of all Reals, such that the > nth digit of the nth item can be determined.
"Can be determined"? By what, a finite algorithm? Cantor's proof requires on such thing. All it supposes it that the nth digit of the nth item *exists*.
> And exactly where does the bit about a "finite algorithm" appear in > Cantor's original proof?
Nowhere at all, which is exactly the point you keep missing! *Your* proof of computability of the antidiagonal requires a finite algorithm. Cantor's proof uses no such assumption.
> Cantor asks for a list of Reals - defined in advance
No, nothing need be "defined in advance". The proof is that if any mathematical object happens to be a list of reals, then it fails to be a complete list of reals.
