In article <email@example.com>, firstname.lastname@example.org 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?
How about the real interval from sqrt(2) to sqrt(3). Since at least countably many irrationals occur as square roots of rationals, not to mention cube roots and higher order roots, there are quite enough such intervals, too.
> Take the well-ordered set of all undefinable > real numbers, then remove 10 elements.
There are no totally undefineable real numbers, only individually undefineable ones, uncountably many of them in every open real interval of strictly positive length. --