Re: Cantor's diagonal proof wrong?

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 11/14/04 

Date: Sun, 14 Nov 2004 14:17:25 -0700

In article <20041114145335.388$26@newsreader.com>,
 curt@kcwc.com (Curt Welch) wrote:

> Chairman of the David Hilbert Appreciation Society
> <mathgeekxxiiii@hotmail.com> wrote:
> > Curt Welch wrote:
>
> > When we diagonalize we get an infinite length string...
> >
> > Think about this please, don't just react.
>
> Yeah, as I just posted before I read your post, I saw that argument coming.
> Or, from another perspective, what everyone else was saying sank in before
> I read your post.
>
> > Your mistake is common. The problem that most people seem to
> > have with Cantor's well known diagonal proof is that they
> > don't know what the real numbers are.
>
> Yes, that's it. I currently have a different view of what real numbers are
> than what everyone else is using. (as well as a different view of what
> integers or natural numbers are). To continue this investigation, I have
> to master the justification mathematicians use to define both.
>
> When I first started seeing this contradiction develop, I was torn between
> trying to understand if math was based on a contradiction, or if it was
> just a matter of choice that the axioms used to define math created a
> contradiction that was easy to avoid of you simply choose different axioms.
> After some debates many months ago (not here), I concluded that it was just
> a matter of choice of which axioms you picked to build your definition of
> math on. But I thought this "proof" that not all integers are in the table
> of integers made it clear there was a fundamental contradiction in the
> axioms. But it does not.
>
> Or, if it does, I don't know enough about how axioms are used to define
> math to argue the point. So before I can finish answering my own question
> about whether math is based on a contradiction, or not, I have to master a
> deeper understanding of math.
>
> The reason I've looked into this is because various arguments such as
> Gödel's incompleteness theorem spills over into arguments about building
> intelligent machines. And some would claim there is an important
> relationship there that seems to indicate there are limits to what machines
> can do - some even take it so far to believe that it "proves" you can not
> build a conscious machine.
>
> My belief is that you can build a conscious machine - that we, in all our
> glory, are nothing more than a complex biological machine. If this is
> true, then everything we create, like all our ideas about math, are nothing
> more than the product of our behavior - which is nothing more than the
> actions of a complex physical machine.
>
> At the heart of all this is the issue of how the mental world ties to the
> physical world. And that touches on how the ideas developed in all the
> fields of reason, like math, relate to the physical world.
>
> The fact that it's reasonable in math, to talk about the size of one
> infinite set being larger than the size of another infinite set, doesn't
> seem to tie correctly to the physical world.
>
> So if the world of math is nothing more than something happening in the
> physical world, why would the two worlds not seem to "fit together" better?
> This is the ultimate question I'm exploring.
>
> I believe the answer is that the "world of math" which we have created, is
> fundamentally different from the physical world. We have used the power of
> language to create a world of math, which can not exist in the physical
> world.
>
> I think hidden in here is a subtle, but important, distinction between the
> ideas of "existence" and "describing existence" which allows us to define
> some really crazy things. We know for example it's trivial to use language
> to tell a lie - to make up a story which we know is not real. But, just
> because a story is not real, doesn't mean it can't be true at some place,
> and some time, in the universe.
>
> But, is there a point, where you use language to create a story, which can
> never be true in this universe? I think there is, and I think the world of
> math has done just that. They have used language to create an imaginary
> world, and at some point, they cross the line from possible, to impossible,
> in the physical universe. And I think that point happens when we pretend
> that infinite sized things, like the real value 1/9, can be constructed in
> zero time, and that we can then manipulate this value as if it existed.
>
> It's the confusion between manipulating the word which represents an
> object, vs. manipulating the object itself. Once you start using words to
> describe other words (which is what math is all about), the distinction
> between words and objects gets lost. And at that point, I think we start
> to define things with words, that can never exist as objects in the
> physical universe.
>
> So, this could mean, that the idea of different sized infinite sets is an
> idea that can not exist for real in the physical universe, even though we
> have no problem using language to create an imaginary world where these
> things do exist.

There is no such thing in the physical world as even a natural number.
"One", "two", "three", etc. are all entirely conceptual, not physical.
If you insist on physicality, give up mathematics.
>
> So, my focus, is trying to understand where language allows us to leave
> physical realty behind, and exactly what happens when we do that. But I
> see I need to learn more about the language of math before I can better
> understand what we have defined here.

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