Am Mittwoch, 11. Dezember 2013 14:07:50 UTC+1 schrieb wpih...@gmail.com: > On Wednesday, December 11, 2013 4:23:18 AM UTC-4, WM wrote: > > > > > Of course the set of all finite words is listable. This list is constructable. The set of all constructable numbers is a sub list. > > > > However, this sub list is not constructable. (Thus to a constructivist > > the collection of constructable numbers is not listable. You fail to realize > > that to a constructivist, the fact that a collection G has the property > > that every member of G is an element of K does not make G a subset of K.) > > Your frequent reference to a non-constructable list should give you pause. > > > > William Hughes >
The misunderstanding happens because an imprecise use of words like "set" or "list", I think.
It's clear that there is a constructible list of all finite words. And it is also clear that this list of finite words should comprise all constructable numbers. Now, in the frame of e.g. ZFC it is not possible to extract these constructible numbers out of the list of all finite words. So, in the sense of ZFC, there is no set of all constructible numbers.
Set theory fails to detect that there have to be "more" finite words than constructible numbers. But we can know it in spite of that fact.