Re: Cantor's diagonal proof wrong?

From: Reinhard (reinhard.neuwirth_at_optus.com.au)
Date: 11/28/04 

Date: 28 Nov 2004 01:52:43 -0800

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

> > 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

>
> > >
> > > 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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