On Apr 11, 8:49 pm, Quadibloc <jsav...@ecn.ab.ca> wrote: > > However, there is a set known to have cardinality aleph-1, the set of > well-orderings of the integers.
Well, sort of. Actually, the set of well-orderings of the integers has the cardinality of the continuum, which may or may not equal aleph_1. It's the set of *order types* of well-orderings of the integers that absolutely has cardinality aleph_1. That is, you definite an equivalence relation on that set of well-orderings, two orderings being called equivalent just in case they are isomorphic, and the the resulting equivalence classes are aleph_1 in number. (That's what you meant, but mathematicians make a big deal of saying what you mean and meaning what you say.)