On Thursday, 27 February 2014 01:48:53 UTC+1, Virgil wrote: > In article <firstname.lastname@example.org>, > > email@example.com wrote: > > > > > > > > > > You believe that all real numbers R except a countable set D are > > > undefinable. >
> The reals are COLLECTIVELY defined
That does not make them enumerable.
> 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.
Therefore they cannot be enumerated other than collectively by " 1 set". > > > > Similarly, Given a set of naturals numbers ther must be subsets of that > > set which cannot have finite definitions because there are must be more > > such subsets than there can be finite definitions of them.
But that is not of interest and does not contradict my claim: The elements of R \ D are undefinable and therefore not enumerable.