Re: Cantorian pseudomathematics

david petry wrote:
> Keith Ramsay wrote:
> > |> Some results like Cantor's theorem are perfectly natural for
> > |> a constructive mathematician like Bishop.
> > |
> > |You might say that, but the constructive version of Cantor's theorem
> > is
> > |not the same theorem as the classical version.
> >
> > I don't know in what relevant sense you consider them not
> > the same.
>
> I know we've discussed this a few times before. The result that is not
> disputed by either constructivists or classical mathematicians is that
> if we are given a well defined list of well defined real numbers (so
> that every digit of every number can be computed), then the diagonal
> method gives us a new number not on that list. The classical
> mathematicians claim, essentially, that the argument is still valid
> when the list and the numbers on the list may not be well defined.

Cantor's theorem is that no set is equinumerous with its power set. As
to the diagonal proof of the uncountability of the reals, no matter
what you mean by "well defined", of course the argument still holds, as
it holds a fortiori, since it makes no assumption about the enumeration
other than it is an enumeration of all real numbers (or subsets of
omega, or whatever) and no assumption about the listed reals other than
that they are reals. Moreover, the proof does not require making a
reductio ad absurdum assumption that all reals are listed, but rather
we can just as well show that any 1-1 mapping from the set of natural
numbers to the set of reals (or the power set of omega) is not onto the
set of real numbers (or the power set of omega).

MoeBlee

.

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