"Tim Little" <tim@little-possums.net> wrote in message news:slrni1m63n.jrj.tim@soprano.little-possums.net... > On 2010-06-18, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: >> I made no premise. > > Sure you did: you assumed that no list of computable numbers can > exist. You also assumed an incorrect definition of "computable". >
No, I assumed that a list of all computable numbers can exist. Then I gave a simple algorithm which forms a computable number which is not on the list. I therefore proved that no list of all computable numbers can exist.
It is *exactly* the same as Cantor's proof that the Reals cannot be listed.
It is of interest because it is known that the computable numbers are countable. Therefore the property "cannot be listed" is *not* the same as the property "is uncountable".
Cantor's diagonal proof does *not* show the Reals are uncountable; it just proves the much weaker statement that "the Reals cannot be listed".
