There is no set whose cardinality is strictly between that of the integers and the real numbers.
Independence of ZF and ZFC
In 1940, Kurt Gödel showed that it is impossible to disprove the Continuum Hypothesis (CH for short) in ZF with or without the Axiom of Choice (ZFC).
In 1963, Paul Cohen showed that it is impossible to prove CH in ZF or ZFC.
These results together show that CH is independent of both ZF and ZFC.
This problem is no. $1$ in the Hilbert 23.