Continuum Hypothesis

Hypothesis
There is no set whose cardinality is strictly between that of the integers and the real numbers. Symbolically, the Continuum Hypothesis asserts that $\aleph_1 = \mathfrak c$.

Independence of ZF and ZFC
In $1940$, showed that it is impossible to disprove the Continuum Hypothesis (CH for short) in ZF with or without the Axiom of Choice (ZFC).

In $1963$, 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.