Re: Cantor's diagonal proof wrong?

From: Curt Welch (curt_at_kcwc.com)
Date: 11/28/04 

Date: 28 Nov 2004 04:04:42 GMT

reinhard.neuwirth@optus.com.au (Reinhard) wrote:
> curt@kcwc.com (Curt Welch) wrote in message
> news:<20041114013915.877$0a@newsreader.com>...

> > So you just reverse the digits in the integer to create the real. I
> > claim this mapping is one to one and covers all the reals in that
> > range. For any real you give me, I can easily give you the matching
> > integer.
>
> *************************************************************************
> *****
> *************************************************************************
> ***** Curt, This should be an easy one for you, then.

You are about 300 messages too late. :)

> I give you the
> (irrational) real 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14... which is
> constructed by concatanating all the integers after the decimal point,
> ad infinitem. I firstly appeal to you to recognise that the length of
> that (irrational) real is the kind of infinity you said you are
> familiar with and which you accept. Secondly I appeal to you to
> recognise that this is indeed an irrational, the recipe (algorithm) of
> its construction ensures there is no repetition anywhere after the
> decimal point (this can be proven rigorously but I don't think we need
> to bother). Being an irrational I would now kindly ask you to write
> this real down in your reverse fashion.

I just write it backwards following your same algorithm. Why do you think
it's important to right left to right? :)

> What is the first digit?

Well, that's what was made obvious to me. Integers are finite, and reals
aren't. So even though I can "write" the natrual number backwards just as
easy and just as fast as you can "write" the real, what I end up trying to
write is not a natural number. And that's where my argument falls flat.

But even beyond that, I later tried another variation of the argument to
remove the definition of natrual numbers and reals from the picture. But
that in the end doesn't work either.

The reason my argument doesn't work is becuse I was using properties of the
physical world to debate the validity of the proof. But I later came to
see that mathematics is a study of the properties of language, and not a
study of the properties of the physical world. And, within the domain of
language, Cantor's proof is consistent with all the other words it is based
on. And consistency of the language is all that is required to make it a
valid argument in mathematics. So even though it seems to take us to a
place that is inconsistent with what we know about the universe, that's
just not important. To prove Cantor wrong in the scope of mathematics, we
must find an inconsistency in the words. And that is not so easy to do. 

-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/