In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> Am Dienstag, 10. Dezember 2013 21:39:31 UTC+1 schrieb wpih...@gmail.com: > > > > > > Your frequent claim is that the set of definitions of numbers (and hence > > the set of constructable numbers) can be listed. > > My claim is that this set is potentially infinite.
That would mean that it cannot be listed, since a potentially infinite set is not completeable, and a list must be.
> If set theory with its > actual infinity is correct, then this set can only be countable.
But if set theory is correct there are uncountable sets. > > > > The fact that this list must be non-constructable should give you pause. > > > Of course the set of all finite words is listable.
Then it must be either finite or actually infinite, as your potentially infinite sets are not listable.
> This list is > constructable.
Then it is either finite or actually infinite because your ambiguously potentially but not actually infinite sets are not listable.
Being listable or constructable requires that no members, even if there would have to be infinitely many of them, can be unlisted or not actually constructed, thus not actual.