In article <email@example.com>, firstname.lastname@example.org wrote: > On Thursday, 27 February 2014 10:00:55 UTC+1, Virgil wrote: > > In article <email@example.com>, > > firstname.lastname@example.org wrote: > > > On Thursday, 27 February 2014 01:48:53 UTC+1, Virgil wrote: > > > > In article <email@example.com>, > > > > firstname.lastname@example.org wrote: > > > > > You believe that all real numbers R except a countable set D are > > > > > undefinable.
> > WRONG!
> You have said so at least.
That is a lie. I have been very careful not to say that, because, as stated, it is not true. The set of all reals is definable, so ther are no reals which are totally undefinable, they are all collectively definable but most of them are without any individual definition.
> > > > The reals are COLLECTIVELY defined
> > > That does not make them enumerable.
> > In fact, their verydefinition makes them non-numerable.
> It makes them non-definable.
Then how does WM manage to speak of them at all?
> > > > by the standard definition of the field of real numbers, but > > > > that collective definition does not guarantee that each of them > > > > also has an individual definition separable from other > > > > individual definitions of all other real numbers.
> > > > Similarly, Given a set of naturals numbers ther must be subsets > > > > of that set which cannot have finite definitions, unless WM can > > > > drum up more thatn countably many definitions.
Until WM can drum up some set which is equinumerous with its power set, there will be sets not all of whose subsets can be named, and sets not all of whose members can be named. --