Re: Cantor's diagonal proof wrong?

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 11/29/04 

Date: 29 Nov 2004 05:00:55 -0800

reinhard.neuwirth@optus.com.au (Reinhard) wrote in message news:<ce3c181d.0411280152.6200ea1c@posting.google.com>...
> mueckenh@rz.fh-augsburg.de (W. Mueckenheim) wrote in message news:<fb701d3c.0411271326.475476c0@posting.google.com>...
> > reinhard.neuwirth@optus.com.au (Reinhard) wrote in message news:<ce3c181d.0411262138.7d988d6@posting.google.com>...
> > > curt@kcwc.com (Curt Welch) wrote in message news:<20041114013915.877$0a@newsreader.com>...
> >
> > Curt, This should be an easy one for you, then. I give you the
> > > (irrational) real 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14... which is
> > > constructed by concatanating all the integers after the decimal point,
> > > ad infinitem. I firstly appeal to you to recognise that the length of
> > > that (irrational) real is the kind of infinity you said you are
> > > familiar with and which you accept. Secondly I appeal to you to
> > > recognise that this is indeed an irrational, the recipe (algorithm) of
> > > its construction ensures there is no repetition anywhere after the
> > > decimal point (this can be proven rigorously but I don't think we need
> > > to bother). Being an irrational I would now kindly ask you to write
> > > this real down in your reverse fashion. What is the first digit?
> > > Another example, there are algorithms for the construction of Pi =
> > > 3.14159... ad infinitum, non repeating, truly irrational, but cannot
> > > be as easily proven as the number stated above.
> >
> > Excuse me, could you tell me whether the digit number 10^10^100 of pi
> > (or of your number given above) is an even one or an odd one? Do you
> > believe such a number to exist, although no one will ever be able to
> > answer this question?
>
> With 0.123456789101112131415... I may possibly be able to tell you if
> 10^Googool is even or odd - after 30 minutes I will either know, or be
> confident that after another 30 minutes I will, or give up, but with
> Pi, no, cannot tell you. However, I understand modern numerical
> methods compute the digits of Pi by doubling the number of digits
> every pass, so it's not a matter of getting the job done one by one,
> for which there is not enough time, and I therefore believe there may
> be hope for answering your question. At any rate, I certainly believe
> that the digits exist and in both cases are one of the set
> {0,1,2,3,4,5,6,7,8,9}, hence are either odd or even with an even
> chance to be either. All of the above, except that it can ever be
> computed, I believe of the digits number Googool ^ Googool. Sorry,
> what was your question?
> RN
>
It is not a matter of time. You could employ your children and
grandchildren. It is a matter of hardware. The whole universe
(accessible to us) has less than 10^80 protons and surely less than
10^100 particles, each being capable of storing 1 bit.
WM

> > > At one
> > > stage I grasped 'infinity' as a serious of steps, which I could choose
> > > to let never end (as a mind game), and I have since been perfectly at
> > > ease with Peano's axioms and inductive reasoning,
> >
> > That is the so-called potential infinity, which marks a direction, but
> > never causes us to leave the finite domain.
>
> Agreed
> RN
>
> >
> >
> > > > Lets create a table of integers like this:
> > > >
> > > > ...000000
> > > > ...000001
> > > > ...000002
> > > >
> > > > ...000010
> > > >
> > > > ...000123
> > > >
> > > > It's just a normal list of integers, but instead of following the normal
> > > > convention of leaving off the leading zeros (which we all know are implied
> > > > even if we don't write them) I include them in that table.
> >
> > Your ideas are perfectly reasonable, if you add the remark that an
> > infinite sequence of digits (which would not form a natural number)
> > could only emerge from a list with at least one line enumerated by
> > omega or infinity. All other numbers are natural numbers and, hence,
> > are finite. But in a line enumerated by omega, also Cantor could not
> > find and exchange the diagonal element. Thís very idea of the diagonal
> > of the naturals can be found in my preprint
> > arxiv.org/abs/math.GM/0305310
>
> Why can I not have an infinite sequence of digits in second place,
> rather than in the place omega? Besides, I can understand omega (by
> which I presume you mean the ordinal equivalent of the cardinal aleph,
> the size of the set of naturals ?), but I cannot imagine it as
> describing the position of the "last" item in an infinite list.
> RN

Any line enumerated by a natural number has a finite number of lines
as predecessors. Hence, the diagonal of the line numbers 111...111
does exist as a finite, hence natural number.

But here is another puzzle:

As long as n is finite, the sequence {2, 4, 6, ..., 2n} contains
larger numbers than its cardinal number is.
If {2, 4, 6, ..., 2n} does not contain larger numbers than its
cardinal number is, then n cannot be finite.

Regards, WM

>
> >
> > > >
> > > > So lets use Cantor's logic on this table and see if we can construct a
> > > > number which is not in the table. We take the numbers from the diagonal,
> > > > and construct the number ...111111 just like we did above.
> > > >
> > > > Since we construct this number by changing a digit from every row, we know,
> > > > by Cantor's logic, that the resulting number can not be in the table.
> > > > Therefore, with the wisdom of Cantor, I've proved that the number of
> > > > integers is greater than the number of integers. There are some integers
> > > > which are simply not in the list of all integers.
> > > >
> > > > Ok, so if Cantor was wrong, why was he wrong?
> > > >
> > > > The answer is one already well known to mathematicians. They just never
> > > > realized how it applied here. You can't use infinity as if it existed. It
> > > > doesn't exist. "infinity" is only a name for something which can not
> > > > exist.
> >
> > In particular actual infinity, the same as "finished infinity", is
> > obviously a contradiction.
> >
> > > > Has any one else put forth this same argument (or others) that Cantor's
> > > > proof is invalid?
> >
> > Here are some other preprints: arxiv.org/abs/math.GM/0408089
> > arxiv.org/abs/math.GM/0403238
> >
> > Regards, WM

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