Re: Cantor's diagonal proof wrong?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 05:31:52 -0600

On 22 Nov 2004 02:01:07 GMT, curt@kcwc.com (Curt Welch) wrote:

>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.

And now stop pretending to be insane for a second, and tell us
which of the following seems more likely:

(i) The problem is that you simply don't understand how language is
  used in mathematics.

(ii) The problem is that you understand better than anyone else how
  language is used in mathematics.

>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).

No, it's taken you a few days to come up with a new and fancy-sounding
rationalization for your refusal to accept a simple proof of something
you don't find intuitively clear, after seeing various more concrete
disproofs die in flames.

>> [...]
>
>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".

No! You're deaf, stupid or crazy. That axiom does not bring anything
into existence. The fact that that's an axiom has nothing whatever to
do with what "really" exists. It's an _axiom_. Something we _assume_.

>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. :)

************************

David C. Ullrich

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