Re: Cantor's wrong! (doubt it)

From: Proginoskes (proginoskes_at_email.msn.com)
Date: 06/16/04

  • Next message: T.D. Lassagne: "Re: Cantor's wrong! (doubt it)" 
    Date: 15 Jun 2004 19:51:52 -0700

    "Chairman of the Ozzy Osbourne Appreciation Society" <mathgeek42@hotmail.com> wrote in message news:<udqdnSvlVare5FLdRVn-vA@comcast.com>...
    > <remlaps@despammed.com> wrote in message
    > news:2b2845e7.0406151239.6726bfbe@posting.google.com...
    > > Has this web page been accepted or discredited?
    >
    > When someone publishes a "disproof" of a result that's been accepted
    > for the last 100 years it's safe to assume that the author doesn't
    > understand basic principles that are involved in the original proof,
    > and this case is no exception.

    Or they don't know about the proof.

    > This guy has proven, not that the reals are countably infinite but that
    > the infinite set of finite length strings on a finite alphabet are.
    > Which of course has been known for a while (did Cantor prove that
    > as well?) .

    The number of "words" (with letters from a finite alphabet) of length n
    is finite. The number of words is the union of these sets. Since each set
    is finite, and the union is of a countable number of sets, the union itself
    is countable. This more general result was proved by Cantor.

    > > At least now if someone searches groups.google.com for andyland and cantor,
    > > they'll find one person who is skeptical of the validity of this argument.
    > >
    > > Any one else want to support it or dispute it? for the archive?

    It's wrong. In fact, his algorithm will never produce 1/9 (which happens to
    be rational). [Actually, he agrees to this on his webpage.]
         -- Christopher Heckman

    P.S. Here's the e-mail I sent to him: ---------------------------------

    Mr. Shane,

    Please remove your webpage
    http://www.andyland.com/education/rensselaer/01999spring/paradox/cantorswrong.html
    from your website, as it is misleading and obviously incorrect.
         I can provide a real number which your program will never produce: 1/9.
    Your program will only produce numbers which have a finite expansion in base
    10, and since 1/9 isn't one of these, it will never be produced, and so you
    don't have a complete listing of the real numbers, even if your program runs
    forever. (The set of all rational numbers _is_ countable, btw, and you are
    producing a subset of these.) If you can't give me a value of N which 1/9
    is bound to, then your program cannot generate _all_ real numbers, since I
    found (at least) one which isn't.
         In fact, you can use Cantor's argument to deduce that 0.211111... will
    not show up in the list. The kth decimal place of the kth number is 1 if
    k = 1, (0.1 is bound to 1) and 0 for any larger k (since the fraction
    terminates before the kth place), and I can prove this for any k, without
    writing out the complete list.
         On your webpage, you say:

    > In particular, the set of lengths at each point in time is the set of
    > natural numbers. ...
    > Then there is the objection that my algorithm may hit 1/3, as shown, but it
    > will never hit any infinitely long irrational numbers. The answer to this is
    > simply that my algorithm clearly does not discriminate against any type of
    > number, it hits them all.

    Your algorithm, however, must produce _all_ real numbers, in order for what
    you claim to be true. These two sentences contradict each other.
         Your webpage is full of contradictions like this. If you read it
    carefully, you will find lots of reasons why your program won't work, and in
    fact, no such program will.

    > The diagonalization proof presupposes that, at least in theory, there is a
    > point in time when we can stop and have all of the real numbers from 0 to 1
    > listed.

    Actually, it only presupposes that you have a 1-1 correspondence. It will of
    course be impossible to write this out, since it will take an infinite amount
    of time to write it out, but Cantor's proof does not require that you do
    write it out. If you use this "there is a point in time when we ... have all
    real numbers" argument, you can also show that infinite sets
    do not exist.
         -- Christopher Heckman
            Ph.D. Algorithms, Combinatorics, and Optimization
            Department of Mathematics and Statistics
            Arizona State University

  • Next message: T.D. Lassagne: "Re: Cantor's wrong! (doubt it)"

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