[jfarmer]

Whether a set of cardinality strictly between the rationals and reals exists is independent of ZFC.

https://en.m.wikipedia.org/wiki/Continuum_hypothesis

There are many sets which are strict supersets of the rationals and strictly sheets of the reals, of course.

[jfarmer]

"strictly subsets of the reals", thanks autocorrect.

For example, if ℚ is the rationals and ℝ is the reals then ℚ∪{√2} is a strict superset of the rationals but a strict subset of the reals. However, it still has the same cardinality as the rationals (cf. https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...)