The purpose of the question is to understand why the number $\sqrt[3]{2}$, that was proven rigorously by ancient Greek is an impossible number (even at infinity), by their three famous impossibility problems,
this one was simply the impossibility of doubling the cube,
But, much later and oddly after many centuries, particularly (in the middle ages), mathematicians fond $\sqrt[3]{2}$ also at infinity, and they widely accepted it as a real number, which would imply the invalidity of the most famous Ancient Greek rigorous proof!
Of course it so clear to even a school student that $$a^3 = 2b^3$$ in positive constructible numbers is impossible equation, thus $\sqrt[3]{2}$ is also impossible number as a result, this was the basis of Greek famous rigorous proof
I had chosen this number $\sqrt[3]{2}$ in particular since it was the oldest famous number that encountered the Greek with a tremendous challenge that is still valid!