Re: Another stab at Cantor

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 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.

Mark Adkins
msadkins04@xxxxxxxxx

Such a binary string which eventually becomes repeating is a rational.
Similarly for any base, eventually repeating strings can always be
expressed as fractions. E.g., decimal 0.142857142857... = 1/7
.

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