After Cohen proved the CH (Continuum Hypothesis) was independent of ZFC, most efforts on the topic seemed (by my reading) to be along the lines of either philosophical arguments for the CH being true or attempts to derive axioms that were equivalent to the CH being true.

But what if we assumed the CH were false, and then, parallel-postulate style, derived what mathematics would happen as a result? Has anyone done this? Has anyone failed at this? Does anyone have any idea of what set with a cardinality between the natural numbers and the real numbers would look like, even in a wildly hypothetical sense?