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?


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