Well done.
Has anyone named a set in between the rationals and the reals?
Has anyone named a set in between the rationals and the reals?
You can kinda think of it as "the set of real numbers MOD the set of rationals". Kinda. And they're dorked up because the length is infinitesimal, but a countably infinite number of them add up to length 1.
For a more precise explanation see here: https://math.stackexchange.com/a/137959/287133
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.
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...)
Assume that CH fails and well order the reals. An initial segment of length omega_1 is an example of such a set.