Re: Cantor's diagonal proof wrong?

From: robert j. kolker (nowhere_at_nowhere.net)
Date: 11/22/04 

Date: Sun, 21 Nov 2004 22:37:10 -0500

Curt Welch wrote:

>
> But in fact, the problem is buried deep down in the language. It just
> doesn't produce any differences until you start to play infinity in the
> right (or wrong) way. So it's taken me some time to trace the
> contradiction down to the definition of math (which actually is where from
> other things I already knew it was but it's just taken me a few days to see
> the obvious).

What contradiction? All of the known anomalies have been removed from
set theory.
>
> I think that alone shows where problems starts. It's the understanding of
> what "exists" means. In my universe, NOTHING can EXIST without a processes
> being involved to bring it into existence. And the key side effect of that
> is that creation always takes time. So the fall out of that is that
> infinite amounts of creation will always take infinite amounts of time, and
> never complete.

You are confusing cause with existence. One can deal with existence
independent of constructive algorithms. It is one all the time in modern
mathematics. A mathematical theory does not have to be empirically
correct. The only requirement is logical consistency.
>
> In the language used to define math, the concept is that if you can define
> it, it just exists, and that's all there is to creation. That alone I
> think explains the differerence between my view, and the standard
> mathematical view.

Which is why your view is totally unsuitable for mathematical discourse.
If you want to do physics then do physics.
>
>
>>All I did is write down a formula.
>
>
> Which is exactly how it works. If you can write down the formula, it
> exists in the world of math, and that's all there is to creation.

Admirably correct. So what is the problem?
>
> So to define the set of all natural numbers, all you have to do is write
> down the axiom of infinity:
>
> There exists a set X that contains the empty set {} and for
> every set Y that belongs to X the set Y+1 constructed as
> Y U {Y} also belongs to X.

>
> And with a snap of the finger, the infinite set is brought into existence
> in finite time just because the sentence above started with "There exists".

Nothing has been "brought" anywhere. The notion is Platonic so it is
assumed always to have existed. Mathematics is Platonic down to the
sub-basement level. If that bothers you, then don't do mathematics.
>
> And that's what just can't happen in my universe. In my universe, you can
> construct as much of it as you want in finite time, but you can't construct
> all of it because that would take infinite time.

In that case you can't do modern mathematics. Give it up. The
mathematical community will not hobble itself with your restrictions and
there is no reason why it should.

Bob Kolker

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