Re: Another stab at Cantor

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


Maybe someone already pointed out something like this, but here's an example of one possible ordering of the repeating decimals where your procedure clearly excludes a particular non-repeating decimal. Take the first two repeating decimals to be:

111111111111...
011111111111...

Then every one of the D's that you produce in your process must have a repeated 0 somewhere. However, it is obvious that there are non-repeating strings with no repeated 0's. For example:

011011101111011111 ...
.

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