Re: Cantor's diagonal proof wrong?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 11/21/04 

Date: Sun, 21 Nov 2004 06:05:44 -0600

On 20 Nov 2004 20:29:22 GMT, curt@kcwc.com (Curt Welch) wrote:

>stephen@nomail.com wrote:
>> Curt Welch <curt@kcwc.com> wrote:
>> : amorgan@xenon.Stanford.EDU (Alan Morgan) wrote:
>>
>> :> And after you are done answering that, does 2/3 terminate or not?
>> :> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
>>
>> : I posted a longer article in this thread earlier today in an attempt to
>> : get the silly questions (and answers) back to the big picture of why on
>> : earth I was talking about procedures that do or do not terminate. You
>> : should read that if you want to understand why I was talking about
>> : these things.
>>
>> : I believe math was created as a language for talking about what can,
>> : and can not do, and know, about procedures. But somewhere, it feels to
>> : me, like it went off track and made up some rules of logic which
>> : violate the laws of nature (the laws of procedures). Did it go off
>> : track or not? Is it a bad thing even if it did? I don't know for
>> : sure.
>>
>> Why do you believe this, and why do you think if went off track?
>> You have not actually given any evidence for your "laws of nature"
>> or that something has gone off track?
>
>Yeah, that's a good question to ask. It's the question I've hard a hard
>time finding a way to answer. I've put forth a lot of evidence, but I've
>not put forth a convincing argument for anyone yet.
>
>But in the my previous post today: <20041120132308.395$4f@newsreader.com>
>
>I present an argument that I hope some people will see shows how the logic
>of the diagonal argument has problems. This counter argument doesn't
>depend on the definition of numbers, or reals, or anything said in set
>theory. It simply shows that any argument that takes the form of the
>diagonal argument to prove that a value can not be in an infinite table is
>invalid.

No, it does not show any such thing.

At _most_ it "shows" something of the form "if you look at it that
way then the proof is invalid", which is simply irrelevant, because
the way you suggest people look at it is simply not consistent with
what the words in the statement of the theorem _mean_. _If_ one
interprets "the reals are uncountable" as meaning "the moon is
made of green cheese" then "the reals are uncountable" does indeed
become false, but this is simply silly because of that "if" -
"the reals are uncountable" does _not_ mean "the moon is made
of green cheese".

>Maybe this will help some people open their minds to the
>possibility that what they are looking at is just an illusion. That there
>might be more hidden behind the language than they every suspected.
>
>The trick to this problem is that it's a matter of faith. Like believing
>in God, once you accept it as fact, it's hard to see anything else. You
>build a huge set of defensive arguments to support the initial belief, and
>any single piece of evidence presented is easily ignored against the
>fortress of defense built to support the first belief. You can't even see
>the other position until you are first willing to, if only for a second,
>open a crack in your defenses and look with new eyes at the evidence - even
>though it seems to contradict everything you believe in.

Find a mirror somewhere. This is a precise description of what the
rest of us see you doing: You started with an explanation why the
proof was wrong. You eventually agreed that your initial explanation
was totally bogus. But that doesn't seem to have had any effect on
your conviction that the proof is wrong - you just continue to invent
new explanations, gradually becoming more and more vague and less
relevant to what the theorem actually _says_.

>> [...]
>
>What we do know however, is that any infinite processes you specify will
>require an infinite amount of time and energy, and will never complete.

And this has no relevance whatever, because the statement of the
theorem has nothing to do with "processes".

>> In any case, I thought you were interested in AI, and that you were
>> going to solve the AI problem?
>
>Yeah, I am. And BTW, I don't need to shed light on this area of math in
>order to solve AI. This is not something I must resolve to finish my AI
>work. It's just something that because of my AI work, I was able to spot a
>problem with. So I became curious to further understand the nature of this
>problem I spotted.
>
>> I would expect a successful AI
>> to understand Cantor's proof. I would also expect a successful
>> AI to understand why the diagonal argument does not apply to
>> the natural numbers or the rational numbers. The fact that
>> you do not understand Cantor's proof does not mean that your
>> AI should not.
>
>I understand Cantor's proof.

If you think that the statement above about completing infinite
processes has some relevance then you don't even understand the
_statement_ of the theorem, much less the proof.

>It's trivial to understand. What's much
>harder to understand is why it's an invalid proof. You might remember I
>started this thread by explaining I used to believe the proof. I was
>taught the proof some 25 years ago in school and thought it was a very cool
>proof and understood the logic instantly even though the results were
>surprising. I spent the next 25 years believing it was an obviously valid
>proof and when I've seen the same proof used in other fields like computer
>science, I instantly believed the results of those proofs. I no longer
>believe that. I know things know that I did not know 25 years ago. I now
>know what we are and why we do the things we do. I know what language is
>now.
>
>My AI would have no problem believing that Cantor's proof is valid just
>like I did and just like you do. It would have no problem learning to talk
>just like you do. If it were educated the same way you were, it too would
>be posting the type of messages you post in order to understand what this
>fool named Curt thought he was talking about. But with enough of the
>proper education, it would also be able to understand what I was talking
>about.

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

David C. Ullrich

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