How did Georg Cantor demonstrate that some infinities are larger than others?

Georg Cantor revolutionized mathematics by demonstrating that infinity is not a single, uniform concept, but rather a hierarchy of different sizes.12

He established this by comparing the cardinality of infinite sets—essentially determining if elements of two sets could be placed into a one-to-one correspondence. While he found that some infinite sets, such as the rational numbers, have the same cardinality as the integers, he proved that others, like the real numbers, are strictly larger and cannot be paired off with the integers.12

Cantor’s primary methods included:
* Diagonalization Argument: A technique used to prove that the real numbers are uncountable, meaning they represent a larger infinity than the counting numbers.13
* Power Set Theorem: Cantor proved that the power set (the collection of all subsets) of any set always has a strictly greater cardinality than the original set.12

This led to the discovery of transfinite numbers, such as aleph-null, creating an infinite hierarchy of increasingly larger cardinalities.12