Re: Another stab at Cantor
- From: "William Hughes" <wpihughes@xxxxxxxxxxx>
- Date: 18 Sep 2006 11:06:41 -0700
msadkins04@xxxxxxxxx wrote:
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. L_Omega therefore contains all
non-repeating infinite binary strings, which is to say all irrationals.
All attempts of the form
<An arbitrarily complex way of comming up with a list
L_c>
are doomed. Either
L_c is not a list
or
L_c does not contain its own diagonal.
In either case L_c is not a list of all real numbers.
If your complex way of producing L_c is messy enough, it will not
be clear what charaterizes the elements of L_c. and why the diagonal
is not a member of L_c. It may well be that no one will take the
trouble to find out. However, the fact that no list contains its own
diagonal means that no list contains all real numbers
So you should not be surprised if you say L_c is a list containing all
real
numbers and someone says "no it isn't", without knowing how
L_c was formed. It is not necessary to know how L_c was formed to
know that it is not a list containing all real numbers.
-William Hughes
Mark Adkins
msadkins04@xxxxxxxxx
.
- References:
- Another stab at Cantor
- From: msadkins04
- Another stab at Cantor
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