On Dec 13, 9:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > It is nonsensical because the same could be assumed for Cantor's > diagonal. It would be undefinable and it would be impossible to prove > that it differs from all lines of the list - in particular if > undefinable reals exist and are members of the list. > Of course the diagonal in some cases can be non-definable, that is well known. For example take the set of all definable reals (which is countable), take any bijective function F between the set of all definable reals and the set N of all naturals, then F must be non- definable, since if F was definable then the antidiagonal on it would a definable real that is not in the list of all definable reals which cannot be. And so the anti-diagonal includes F in the formula defining it, but F is a non definable parameter so the Anti-diagonal is not a parameter free definable and thus as we say non definable.
That doesn't mean that we cannot prove it is different from all reals in the list, on the contrary we don't need parameter free definability in order to determine that the diagonal is different from the reals in the original list, we can do that without it, as Cantor did.
Your error is that you think too much of non definability. It is not so destructive as you think.