On Sunday, May 4, 2014 4:15:03 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > On Sunday, 4 May 2014 01:12:40 UTC+2, Virgil wrote: > > > > What elements has the set of all individually undefinable real numbers? > > All but the countably many real numbers for which there can be finite > > individual definitions. > > > Does it differ from one of its subsets > > It only need differ from its proper subsets, of which there are many. > > E.g., EVERY subset of only those individually undefinable real numbers > > within a real interval having distinct rational endpoints will be a > > proper subset of the set of all individually undefinable real numbers. > > And without rational endpoints? Take the well-ordered set of all undefinable real numbers, then remove 10 elements. How can you do so by applying the axiom of extensionality?
You should look into the Theory of Indescernibles and how they can be added to L without violating any of the Axioms.
For example, let S be the Set of Indesceribles in R. Let G be the Set of all 10-tuples of S. Then, for any H e G, S\H exists.
Now, given two Elements of G, we can't tell them apart. But this doesn't violate any Axiom.