On Sunday, May 4, 2014 2:05:43 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > On Sunday, 4 May 2014 00:32:08 UTC+2, Zeit Geist wrote: > > On Saturday, May 3, 2014 2:04:34 PM UTC-7, muec...@rz.fh-augsburg.de wrote: > > > On Saturday, 3 May 2014 20:55:10 UTC+2, Virgil wrote: > > > > > There is only one such set which has no elements. the empty set, > > > What elements has the set of all individually undefinable real numbers? Does it differ from one of its subsets by applying the axiom of extensionality? > > Depends on the Model you're working in. > > If it is Non-Empty, then it differs from All but One of its Subsets. > > Of course, in L ( Gödel's Constructable Universe ) All Reals are "Definable" in a certain way. > > And yet, they remain Uncountable in L. > > Here we talk about mathematics performed in English. There definable and uncountable exclude each other, not only "in a certain way".
I was referring to the fact that the Property of "Definability" in the Constructible Universe is stated in a particular way, as to avoid the Paradoxes that can result when looking at "Definability" from some Naive Perspective. Like you are doing.
You don't look at how Definable Objects are form in a Hierarchical manner. You also ignore the Definition of Countability in this Context and the Fact that No Definable Bijection Exist between your Set of Definitions and the Natural Numbers.