I was having this question while trying to show that the set of all rational numbers under addition, $(\cal Q,+)$, and the set of all real numbers under addition, $(\cal R,+)$, cannot be isomorphic.
Infinite uncountable set does not have bijective function from the set of natural numbers to it by the very definition of being uncountable set. Hence, it's not possible to find a bijective function between infinite countable set and infinite uncountable set.
This was going to be my reasoning for $\cal (Q,+)$ and $(\cal R,+) $ cannot be isomorphic. Am I approaching the problem with right intuition?