How did Georg Cantor help establish set theory as a branch of mathematics?

Georg Cantor is universally recognized as the founder of modern set theory, a field he established through his pioneering work on the nature of infinity in the late 19th century.12

Cantor’s primary contribution was the systematic study of infinite sets, which he approached by applying the device of one-to-one correspondence to compare the sizes, or cardinalities, of different collections. By doing so, he demonstrated that some infinite sets are larger than others—for instance, proving that the real numbers are uncountable and thus more numerous than the integers.321

These insights led him to develop the concept of transfinite numbers, creating an infinite hierarchy of cardinalities (${\aleph }_0,{\aleph }_1,{\aleph }_2,\dots$). While his work initially faced significant resistance from contemporaries like Leopold Kronecker, it eventually became fundamental to analysis, topology, and the logical foundations of mathematics. Although Cantor did not formalize the axiomatic foundations of the field, his revolutionary ideas regarding the "actual" infinite transformed mathematical thought.123