On 2010-06-17, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: > Cantor's proof applied to computable numbers proves you cannot form > a computable list of computable numbers.
Cantor's construction applied directly to lists of computable numbers shows almost nothing. Given any list of computable reals, you can produce an antidiagonal real not on the list. Unfortunately the construction says nothing about whether the antidiagonal is computable or not.
It takes quite a bit of extra work (not part of Cantor's proof) to deduce that there is no computable list of all computable reals.
> Cantor's proof applied to Reals proves you cannot form a computable > list of Reals.
Incorrect, and even WTF-worthy. Cantor's proof *is* applied to reals, and nothing in it has anything whatsoever to do with computability.
> The property "that you cannot form a computable list" is not the > same as the property "is uncountable". For example, computable > numbers have the first property but not the second.
I agree. It was Herc's deranged ramblings that brought computability into this discussion in the first place, when they really are completely irrelevant to Cantor's proof.
> Cantor's proof is about what can be expressed in a list, and not > directly about uncountable sets (which don't even get mentioned in > the proof).
"Cannot be expressed in a list" is actually one of many equivalent definitions for the term "uncountable". Whether Cantor himself used that specific word is irrelevant - the mathematical content of his proof was to establish uncountability.