In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> Am Dienstag, 3. Dezember 2013 18:53:38 UTC+1 schrieb Zeit Geist:
> > Where in the Axiom Set for the Real Numbers are "Defined Numbers" > > mentioned?
> They are not mentioned because undefined numbers are not numbers.
But there are individually undefineable real numbers numbers that ARE collectively defineable, and thus ARE numbers.
> > Where in you posts have you given a a Formal Conception of "Definable > > Numbers"? > > There is no formal defintion because formal language is incapable of many > necessary things. But every definable number is definable by a finite word > that does not defina another number. In English, a finite word by itself rarely defines anything, though a finite concatenation of words sometimes does. > > > > > > No. ZFC has uncountability. Definable numbers have not.
At least not individually, but the totality of collectively defineable numbers is collectively an uncountable set.
If for no other reason than neither WM nor anyone else, can list them all. And their being listable is the basic requirement for there being only countably many of them.
Or does WM have a definition of a set being countable which does not require its members being listable?
I have never such a definietion outside of WM's wild weird world of WMytheology, and the ones inside of WM's wild weird world of WMytheology make no sense. > > > > > > > > Could you give an example of a Model of ZFC in which all Real Numbers are > > definable and that contains a Bijection between N and R. > > > > Could you give a model of ZFC at all?
There are a lot of such models but they are all incompatable with the nonsense required by WM's wild weird world of WMytheology.
> > > > You should be able to Prove it. > > I did. You seem to have a very bad memeory.
Unlike WM's memory, Zeit Geist's memory does not remember things that have never happened > > Look: Every definition in mathematics is a finite word in English or any > formal language.
Almost every definition in English is not just a world but a sentence, or at least a phrase, having more than one word in it.
> > A subset of a countable set is countable.
BUt supersets of a countable set need not be. And some are not.
Once one allows countably infinite sets, which WM has been doing now for some time, the power set construction will produce uncountable sets from them. --