Re: Cantor's diagonal proof wrong?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 11/16/04 

Date: Tue, 16 Nov 2004 07:05:09 -0600

On 15 Nov 2004 23:46:38 GMT, curt@kcwc.com (Curt Welch) wrote:

>imaginatorium@despammed.com (Brian Chandler) wrote:
>> curt@kcwc.com (Curt Welch) wrote in message
>> news:<20041114231757.327$N2@newsreader.com>...
>> > Virgil <ITSnetNOTcom#virgil@COMCAST.com> wrote:
>> >
>> > > 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.
>> >
>> > I am exploring things that you believe do not exist. And your outlook
>> > is not uncommon in the world. It's by far most common view all of
>> > mankind seems to like to share.
>>
>> I don't think this is true at all. In my experience, if you ask a
>> roomful of non-mathematicians whether i, the square root of -1,
>> "exists" they will mostly claim it doesn't. Ask "Does 3 exist?" and
>> overwhelmingly they respond that yes, it does. Then you can play an
>> amusing little game, trying to establish where their particular
>> boundaries between "existing" and "not actually existing" numbers lie.
>
>Yeah, you are probably right. I have not tried to play that game with
>people.
>
>I'm glad you at least see the issue I'm getting at. People use many words
>like "exist" without trying to understand or justify what they really mean.
>They simply, like parrots, use them correctly like they have been taught.
>Most of what we all do works like that. And for sure, what I write is full
>of that, as you go on to demonstrate below.
>
>> Virgil, of course, or any other mathematician, will just look a bit
>> blank, and say "What do you mean 'Exist'?", but I think this is a
>> minority view. Of course, mathematicians will hedge their response,
>> because in some cases, your "Exist?" will obviously apply to
>> mathematical existence, or not, as in four-sided triangles, even
>> primes greater than 63, and so on.
>>
>> > It's that very fact that makes me at times, believe I've found
>> > something that has been missed for 100's of years. Matematics, by
>> > design, limits it's focus to a scope which does not include the things
>> > I'm investigating.
>>
>> Ah! Two points.
>>
>> a) You have misspelled "it's", which is a pretty serious blunder.
>
>Grammatical and spelling errors get much worse than that in my writing. :)
>
>> b) You've also missed quite a lot that was found 100's of years ago.
>>
>> Several people answered your "proof that the integers cannot be put in
>> 1-1 correspondence with themselves" (or something like that), but I
>> think the answers were not terribly good (a bit Micro$oft-like, if you
>> know the helicopter joke). You have a somewhat ill-defined set of
>> objects, including the integers, 1, 2, 3, 57, 264, etc., and some
>> things like ...11111; you haven't really said quite what else. Which
>> of these is one of your cwintegers (as I'll call them, in honour of
>> yourself!):
>
>Yes, this sounds like fun. What was I thinking an integer was?
>
>Well, for starters, it clearly included the set of all numbers which anyone
>would call an integer. And that includes all the normal well known
>properties of said numbers.
>
>But then I called the infinite string of digits which could be constructed
>from the diagonal of my table, "an integer" and that's where the trouble
>started.
>
>> ...2121212121 * recurring decimals backwards
>> ...5356295141 * fractional part of pi backwards
>> -...1111111111 * cwintegers with a minus sign
>
>Let me just for fun (not to try and fix my "proof"), just try to formalize
>what I was thinking with ...11111
>
>Well, it's the same thing as the infinite series 1*10^0 + 1*10^2 + 1*10^3 +
>....
>
>Now, this is not incompatible with the integers in that you can simply
>substitute 0's for all the leading numbers.

Regardless, you are _changing_ what the word "integer" _means_.
Let's call these things curtigers instead, to prevent confusion.
You have given a correct proof of the utterly obvious fact that
the curtigers are in one-to-one correspondence with the reals
between 0 and 1. But that has no relevance whatever to Cantor's
proof, because he didn't say anything about curtigers, the statement
is about integers.

Oh - I see someone has already named them cwintegers, fine.
Questions about whether cwints "should" count as integers, whether
it's possible to define arithmetic operations on cwints, etc,
have no relevance to the truth of Cantor's theorem. Because words
mean what the definitions say they mean, not what you think
they should mean: even if you convinced everyone that cwints
_should_ be included from now on as integers, they are _not_
included in the definition of the word "integer" as it appears
in Cantor's theorem.

Regarding the irrelevant question of whether it's possible
to make cwints work like integers: Let

  x = ...212121,
  y = ...121212.

Which is larger, x or y? We need to know, because it has some
bearing on what x - y should be.

>> While you are free to finish off the definition, so we know exactly
>> what is and isn't a cwinteger, then you have to start doing some grunt
>> work, proving results that suggest that your cwintegers are useful.
>> The integers used by mathematicians have at least the property that
>> they possess the qualities children learn about informally at a _very_
>> young age. In particular:
>>
>> You can always add, subtract, or multiply two numbers. You can't
>> always divide, but if you can't, it's because there is no answer at
>> all, not because there are lots of answers. You can compare two
>> numbers, and it works: if a>b, b<a; if a<b and b<c, then a<c. Numbers
>> are odd and even, and even+even=even.
>> And lots and lots more. Here are some simple questions about
>> cwintegers you should ask yourself:
>>
>> 1 + 2 = ?
>> 3 - 2 = ?
>
>Those are obvious with cwints because cwints are the same as integers when
>they are finite in size.
>
>> ...999 + 2 = ?
>
>You just substitute ...999 for the correct infinite series, and you have
>your answer. however, the answer can not be translated back into this same
>notation, so a new notation must be created. 1...01. This is 1, with an
>infinite string of leading zeros, and an infinite carry of a 1 value. :)
>
>Now, I have to deal with how everything works with this new notation.
>
>1...01 + 1 = 1...02
>1...01 - 1 = 1...0
>
>1...0 - 1 = ...999
>
>And it continues to get more complex, which I will not work through right
>now. :)
>
>> 1 - 2 = ?
>>
>> Is ...2222 > 5 ?
>
>Yes, clearly.
>
>> Is ...2222 < 5 ?
>
>No, clearly.
>
>But what about ...222 > ...111?
>
>Hard to say what to do about that. But it seems like you could just decide
>that the answer is true. The other option would be to declare it
>undefined.
>
>In either case, you would then have to work though what would either choice
>would mean for all the other operations.
>
>Yeah, clearly, I didn't grasp how much work has been done over 100 years
>ago. :)
>
>> As for all this stuff about AI, I recommend Daniel Dennett's book
>> "Darwin's Dangerous Idea",
>
>Yeah, I've seen reference to it in the past. Thanks for the
>recommendation.
>
>I tend not to read much philosophy. I'm an engineer.
>
>> Anyway, I think a bit of reading would be a good idea, before you
>> announce your Great Discovery.
>
>I find that to be true only to the extent of helping me to communicate with
>others. If the "great Discovery" is a thinking machine, I don't need to
>announce it. I'll let it do the announcing. :)
>
>> > You do not believe the "conceptual" world and the "physical" world are
>> > one in the same. I do. And once you believe that, everything starts
>> > to get very interesting, and everything starts to look very different.
>>
>> But does it make any sense? Do you mean that numbers (and things)
>> Really Exist in the physical universe, for you? Including i?
>
>Yes. But what is "the physical universe" and what does "exist" really
>mean? That's where it gets interesting and that's why I'm able to answer
>the question "yes" when others would like to say you can't answer it that
>way.
>
>In other words, the physical universe I know of is created by the behavior
>of my brain. It's all in my head. It is a purely subjective understanding
>of the universe. But in that virtual creation of my brain, is multiple
>representations of my body, and your body. And in our bodies, are brains,
>which are information processing machines, which create independent virtual
>views of the same physical universe.
>
>So where is the "physical universe" in this loop? Is it that "stuff out
>there" which we normal think about when we use those words, or is, as it
>must be, only the collection of ideas computed by my brain?
>
>The only things that actually exist to us is the electrical activity in our
>brain. If it's not electrical activity in our brain, it doesn't exist to
>us. Everything that exists to us, from our own body, to the body of
>others, to numbers, exist as electrical activity in our brains. Electrical
>activity is the physical movement of electrons. It's physical stuff. It's
>my brain, and all the atoms and electrons in it, changing shape in response
>to its interaction with other matter in the universe. That's what the
>"number 1" is. It's not just an "abstract idea" which exists in some other
>dimension from the "physical stuff". Ideas are the actions of physical
>stuff in this universe.
>
>So, there are many ways to talk about all this, and if you don't follow
>which way I'm talking at the moment, it's easy to get lost.
>
>So, everything we know about, and sense, from our physical sensations of
>the physical world, to all the sensations of the mental world, are created
>by electrons and chemicals flowing in our brains. And the electrons that
>allows us to sense the idea of the imaginary number i, is just as real, as
>the electrons that allows me to sense the keyboard I'm typing on.
>
>The physical world we are able to understand, is not "out there", it's in
>our head, along with the stuff that creates the idea of i. So our
>understanding of the physical world is just as real, as our understanding
>of a concept like i. And all this ability is built out of material from
>the physical world.
>
>> Brian Chandler

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

David C. Ullrich

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