Re: Another stab at Cantor

On 18 Sep 2006 09:50:13 -0700, msadkins04@xxxxxxxxx wrote:

It seems to me that people have correctly pointed out
unimportant errors below - unimportant because they're
easily fixed.

And people have correctly explained why what's below
_must_ be wrong. But I don't recall anyone pointing
out what the actual _error_ below is. Here it is:

Let R be the set of all infinite binary strings which eventually settle
into a repeating digit or pattern of digits. Let L1 be a well-defined
ordering of R. Let D1 be the diagonal number obtained from L1 by the
application of Cantor's diagonal process. D1 is not a member of R.

Let L2 be the list obtained by putting D1 at the head of L1 (that is,
by adding one to the index number of each member of L1, and placing D1
in the first position of the newly indexed list). Let D2 be the
diagonal number obtained from L2 by the application of Cantor's
diagonal process. D2 is not a member of R, and is not D1. Let L3 be
the list obtained by putting D2 at the head of L2.

Let L_Omega be the list defined by the totality of all possible steps
of this procedure. There is no non-repeating infinite binary string
excluded by the procedure.

How do you _know_ that there is no non-repeating string that's
left out? There's no proof of that above - this is just an
empty assertion.

(No, there's no specific string that you intentionally excluded.
But you have to _prove_ that no such string has been excluded.
Good luck with that, btw.)

L_Omega therefore contains all
non-repeating infinite binary strings, which is to say all irrationals.

Mark Adkins
msadkins04@xxxxxxxxx


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

David C. Ullrich
.

Relevant Pages

  • Re: Another stab at Cantor
    ... unimportant errors below - unimportant because they're ... application of Cantor's diagonal process. ... D1 is not a member of R. ... How do you _know_ that there is no non-repeating string that's ...
    (sci.math)
  • Re: Another stab at Cantor
    ... into a repeating digit or pattern of digits. ... D1 is not a member of R. ... There is no non-repeating infinite binary string ... <An arbitrarily complex way of comming up with a list ...
    (sci.math)