Re: Cantor's diagonal proof wrong?

From: George Greene (greeneg_at_cs.unc.edu)
Date: 11/23/04 

Date: 23 Nov 2004 00:12:05 -0800

> "*** T. Winter" <***.Winter@cwi.nl> wrote:
> > Now the interesting thing is that it looks like the diagonal is on
> > the list. But it obviously is not, for the same reason that the
> > infinite set of natural numbers does not contain an infinite number.
> > But CW is thinking that you have to construct initial parts of the
> > diagonal number sequentially, and that is where his statement
> > "every diagonal" comes from.

curt@kcwc.com (Curt Welch) wrote in message news:<20041122132635.076$bc@newsreader.com>...
> Right. You have restored my faith in mainkind. :) You may not know why
> I'm thinking the way I think, but you at least show that you are smart
> enough and have enough of an open mind to understand something new and
> different.

It's not clear that "new & different" was what anybody was
trying, though. We thought we were all talking about all
the OLD natural numbers, the OLD real numbers, and the OLD lists
of them, and the OLD diagonal argument about them.

> And you are right, the argument is exatly the same as my initial argument
> about infinite sized integers but the new form of the argument takes the
> definition of integers out of the picture. I thought taking integers out
> of the argument would be enought to clear up the confusion. But it wasn't.
> If you use the normal mathematical ideas about creation and existence, my
> argument still falls short even though integers have been removed from the
> language. So in the langauge of math, no version of my argument makes
> sense.

It is not a linguistic problem.
There is no such thing as "the language of math". There is a usual
paradigm that is adequate for most purposes but there are mathematical
investigations that go beyond it, for those who care to. There are
infitary mathematical langauge and branches of math that deal with them.
It just so happens you don't NEED anything THAT powerful to figure out
why no set is as big as its powerset.

> Which is why I now say that what I've been talking about is
> actually an issue outside of math.

Hardly.
It is too abstract to be anywhere else.

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