TITLE: In set theories where Continuum Hypothesis is false, what are the new sets?
QUESTION [27 upvotes]: So, say we are working with non-CH mathematics. This means, AFAIK, that there is at least one set $S$ in our non-CH mathematics, whose cardinality is intermediate between $|\mathbb{N}|$ (card. of naturals) and $|\mathbb{R}|=2^\mathbb{N}$, the continuum.
Question: what kind of objects would we find in this set $S$?
Also: is this mathematics radically different from the one where CH holds?
Specifically, are there results that are used in everyday math , at a relatively introductory level, which do not hold on our non-CH math.?. What results that we find in everyday math would not hold in our new math?  Would there, e.g., still exist non-measurable sets? Maybe more specifically: what results depend on the CH?

REPLY [2 votes]: A very natural question for algebraists is the Whitehead problem: is there an abelian group A which is not free but for which Ext^1(A, Z) = 0? Shelah showed that this natural problem was independent of ZFC. If you assume CH is false and Martin's axiom is true, then such a group exists. On the other hand, V=L, which implies CH, also implies such a group does not exist.