On 8 Jan., 17:27, George Greene <gree...@email.unc.edu> wrote: > On Jan 3, 9:19 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > because > > no infinite diagonal of a Cantor list can be defined > > 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!
In this context, a Cantor-list is a sequence of real numbers of the unit interval, in general without repetitions. But that would not disturb the argument.
> IT'S YOUR list! YOU allege that it is well-defined and that > it contains all the reals!
No, I don't, in particular because there is no such thing as "all the reals". Why should I claim such a nonsense?
> But you also allege that there are only countably many reals and > that they are all definable!
I do not allege but can prove that not more than countably many objects can exist a individuals in mathematics and elesewhere.
> In that case, a definable list of all the definable reals IS YOURS > *AND*NOT* > Cantor's!
It seems to be yours. It is neither mine nor 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!
But I can define a list of all finite strings of digits, that is a subset of all rational numbers of the unit interval, where most appear more than once. Here it is: 0.0 0.1 0.00 0.01 0.10 0.11 0.000 ...
> 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!
For instance this list is so well defined, that it cannot be objected that every finite string of the anti-diagonal is an entry of that list. By the way this could also be constructed in decimal. But you may be intelligent enough to understand that? > > However, if the list of all definable reals were ITSELF definable, > then its anti-diagonal > would be definable
Let that strawman rest where it is burried. I am not interested in the question whether suchga list was 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).
This contradiction happens already in the above list of all finite strings which is certainly definable. The anti-diagonal never deviates from every entry (at a finite place - but others do not exist). This contracition shows that it is nonsense to talk about completed infinity.
> Conclusion: the list of all definable reals > IS NOT ITSELF > DEFINABLE! And it most certainly is NOT a CANTOR list!
As I said, definability is not interesting. > > IT'S > > * YOUR * > > DELUSION > You are in error. It is your delusion. I do not believe in completed infinity. Not even in the existence of the list defined above.
> 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!
Do you join me when I say that the list containing all finite strings of 0's and 1's given above does not exist? Do you join me when I say that the notion of countability is nonsense?