On 13 Dez., 11:49, Zuhair <zaljo...@gmail.com> wrote: > 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.
Also the elements of the list could be non-definable, if non-definable real numbers existed.
> 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.
Cantor did not accept non-definable reals. If he had, he would have seen that his proof fails. > > Your error is that you think too much of non definability. It is not > so destructive as you think.
No? Nearly every real number is undefinable. The measure of definable reals is 0. If most reals are non-definable, why must all reals of every Cantor list always be definable? If all reals of the list are definable, then they belong to a countable set. Then we cannot prove uncounatbility. Or can we prove that the set of definable reals is uncountable - because it is countable but there are, somewhere else, undefinable "reals"?