Re: Cantor's diagonal proof wrong?

From: Curt Welch (curt_at_kcwc.com)
Date: 11/22/04 

Date: 22 Nov 2004 02:01:07 GMT

trimble1@optonline.net (Todd Trimble) wrote:
> On 20 Nov 2004, Curt Welch wrote:

> I've said it before; I'll say it again: your objections are
> philosophical, not mathematical. Yes, the problem is *language*;

Yeah, that's right on the money. I see that now. I was confused because
the problem didn't show up in math as a problem until you got all the way
through all the lower level math and started working on this infinite set
issue. It seemed logical to me that if the problem happened here, that's
where I would find the break in logic.

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).

> <Sigh> Well, that's why we have formal languages in mathematics,
> so that such philosophical issues, interesting though they may be,
> are beside the point in deciding validity of mathematical arguments.
> But to argue in an intelligent and informed manner about this,
> one has to understand formal deductions in a logical calculus
> (e.g., Gentzen sequent calculus), and I don't believe you've
> quite come to grips with this (your great strides in the past
> few days notwithstanding).

Yeah, I've seen many web pages referencing all sorts of complex work in the
various types of logical calculus such as the different orders, and more
names and theorems than I can remember. I chuckle when other people post
in this thread that it's all so "simple" that I should be able to master it
in seconds. :) There's lots of language there that I need to master before
I really will understand the history and depth of what has been explored in
the past 100 years.

> One can represent relational (and in particular functional)
> composition by a first-order formula such as
>
> Exists_{y} R(x, y) & S(y, z)

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.

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.

> 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.

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".

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.

The above, is just the description of a procedure that says:

   X = {}
   Y = {}
   loop forever
     Add Y to set X
     Y = Y U {Y}
   next

And procedures like that, even when written as a formula, can never
terminate in my universe. But they are assumed to complete (i.e. exist) in
the universe of math once they are written down.

A quote was posted in the thread where someone famous (Leopold Kronecker?)
once said that God created the natural number and the rest was the work of
man. I'd argue that God created creation and the rest was the failure of
man to understand the work of God. :) 

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

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