> the set of all programs that cannot be proven to halt That's easy: pick the TM which is minimal according to some lexicographic ordering on the specification

> the set of all programs that cannot be proven to halt That's easy: pick the TM which is minimal according to some lexicographic ordering on the specification of TM's. That's not computable (obviously) but it's perfectly well defined. > The set of reals that are not the solution to any equation that can be written with a finite number of symbols Yeah, that one is harder :-) (I would simply say "The set of numbers that cannot be described by any finite number of symbols" in order to short-circuit arguments about what constitutes an "equation" and whether or not Chaitin's constant is the solution to some equation.)