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!
* YOUR *
THAT *YOUR* list of all definable reals is definable, and yoru much deeper delusion that your list of all definable reals lists all the reals!