Re: Another stab at Cantor

In article <1158681712.769259.84930@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
georgie <geo_cant@xxxxxxxxx> wrote:

Arturo Magidin wrote:
In article <1158598213.849000.127730@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<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 such thing. If you think of L_{n+1} as being obtained by
"adding a previous element" to L_n, then L_Omega is not a list: it
does not have a first element.

That would seem to be the desired result. He wants the limiting
case to be the *SET* of real numbers, not the *LIST* of real numbers.

Doesn't matter. This problem is minimal, as has been pointed out time
and time again. His final set of binary strings (there is no need to
introduce the further layer of abstraction of considering them as real
numbers) contains exactly the following strings:


(i) The strings in the original list L1. Call them L1(k), with k
ranging over all positive integers; L1(k) is the string in the k-th
position of the origina list.

(ii) The strings Dk, where D1 is obtained via the diagonal process
from L1, and D(k+1) is obtained via the diagonal process from the
list that has D(k+1-j) in the j-th position, j=1,...,k, and L1(n) in
the (k+n)th position.

And nothing else.

That is his "final" set.

Now, in order to exhibit one (of the infinitely many) binary strings
not in this SET, I consider the following LIST (separate from the
set):

For each positive integer k, if k = 2n is even, then the k-th term in
the list is the string L1(n) which was in the nth position of the list
L1. If k=2n-1 is odd, then the k-th term in the list is the string Dn
described in point (ii) above.

This list contains each and every element in the ->set<-, exactly
once. Let S be the string constructed by applying the diagonal process
to this list. Then S is a binary string, which is not in the SET
described above (the "end result" of the process).


--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
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Arturo Magidin
magidin-at-member-ams-org

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