[longshadow]

"There is a set of numbers 'smaller' than the reals and 'bigger' than the natural numbers" (aka the Continuum Hypothesis) is undecidable....."

Caution must be execised to avoid confusing "undecided" hypotheses from "undecideable" hypotheses; the CH is undecided -- that is, we don't know if it is true or not, and it also makes no difference to standard set theory whether it is true or false (IOW, standard set theory is independent of the CH), but no one to my knowledge has demonstrated the CH can NEVER be decided. IOW, no one has shown the CH is a Gödel statement.

[edsheppa]

You're using terminology in a confusing way. The status of the CH is definitely that it is undecidable. For example look here which I quote in relevant part.

Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the Axiom of choice).

[Doctor Stochastic]

Actually, CH isn't a Goedel statement in that either CH or ~CH can be added to ordinary set theory (Zermel-Franco axiomatization to be more precise) and either both systems are consistent or neither is.

It's analogous to the Axiom of Parallels in Euclidean geometry; one can postulate 0, 1, or many parallel lines can be drawn through point not on a given line; there are thus three (at least) geometries.