Re: Ultimate debunking of Cantor's Theory

WM wrote:

[Peter Webb]
When you do the Cantor trick in base 10, you can prove to yourself that it
always produces a number not on the list. Even if you have 0.500.. somewhere
on the list, you can be certain that you will never get the same number in a
different form, such as 0.49999.. as a result of the construction.

If you could find a single example where the Cantor construction failed to
produce a different number - if for example it generated 0.4999.. when 0.5
was on the list - then you can no longer claim that the Cantor construction
ALWAYS produces a new number.

Such an example is easy to find. Consider the list
0.0
0.1
0.11
0.111
...
and switch 0 to 1 on the diagonal. Then you have at the diagonal the
number 0.111..., but only if this number (with one digit less) is also
in the list.

The guy to whom you are replying has already given an example where
the Cantor construction fails in base 2. However he was writing about
base 10 above, and with a sensible definition of the construction it
is impossible to find an example where the it fails to find a new
number. For example one can define the construction so that the
decimal expansion it gives contains only the digits 4 and 7. The only
way that two different decimal expansions can define the same number
is if one of them contains an infinite string of 9's and the other
contains an infinite string of 0's.

.