One of the most beautiful ideas in mathematics is: some infinities are bigger than others.
Though it sounds strange, it is far from obvious. In the 1870s, Georg Cantor proved that the set of real numbers is larger than the set of natural numbers. No matter how you try to pair them, some real numbers are always left unmatched.
Even more surprising, from any infinite set, we can construct a larger one. In other words:
for every infinity, there exists a greater infinity.
In 1900, David Hilbert praised this discovery, saying:
“Nothing will remove us from the paradise that Cantor has created for us.”