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Topic: Distinguishability argument x Cantor's arguments?
Replies: 15   Last Post: Jan 9, 2013 4:32 PM

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 george Posts: 800 Registered: 8/5/08
Re: Distinguishability argument x Cantor's arguments?
Posted: Jan 8, 2013 11:27 AM

On Jan 3, 9:19 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> because
> no infinite diagonal of a Cantor list can be defined

DANG, you're stupid.
THERE IS *NO*SUCH*THING*AS* "a Cantor list"
IN THIS CONTEXT! Cantor is REFUTING the list!
Cantor is REBUTTING the list! The list comes FROM YOU!
IT'S YOUR list! YOU allege that it is well-defined and that
it contains all the reals!
But you also allege that there are only countably many reals and
that they are all definable!
In that case, a definable list of all the definable reals IS YOURS
*AND*NOT*
Cantor's! If THAT list exists, then THE INFINITE DIAGONAL OF IT
*CAN*BE*AND*IS*
*TRIVIALLY* defined! It is just "the real whose nth place is the nth
place of the nth-defined real ON YOUR, NOT CANTOR'S, list."
It's YOUR list and THAT'S *YOUR* definition of its diagonal!
And if that diagonal can be defined then THE ANTI-diagonal CAN ALSO be
defined!
For ANY DEFINED bit-string, its complement IS WELL-defined!
The nth digit of the complement is 1 - <the nth digit of the original
string>.
If the base is 10 then the nth digit of the anti-diagonal is just 9 -
the nth digit of the diagonal!
THESE ARE TRIVIAL, SIMPLE, STRAIGHTFORWARD definitions!
ANYthing defined this way CAN BE *AND*IS*WELL*- defined!

However, if the list of all definable reals were ITSELF definable,
then its anti-diagonal
would be definable and WE WOULD HAVE A CONTRADICTION that the this
definition
of "the anti-diagonal-of-the-list-of-all-definable-reals" both WAS
definable (since the
above is a definition) AND WAS NOT definable (since it differs from
every row of the
list of definable reals). Conclusion: the list of all definable reals
IS NOT ITSELF
DEFINABLE! And it most certainly is NOT a CANTOR list!

IT'S