How did Cantor’s ideas contribute to the development of modern set theory?

Georg Cantor is universally recognized as the founder of modern set theory. His work fundamentally transformed mathematics by providing a rigorous framework for analyzing infinite sets, a subject previously avoided or considered paradoxical by many mathematicians.12

Cantor’s primary contributions include:

• Cardinality and Comparison: Cantor introduced the concept of cardinality to measure the size of sets. By using one-to-one mappings, he demonstrated that infinite sets could be compared and that some infinities are strictly larger than others.13
• Cantor’s Theorem: He proved that the cardinality of any set is strictly less than the cardinality of its power set. This discovery revealed an endless hierarchy of transfinite numbers, such as aleph-null (${\aleph }_0$) and the continuum ($c$).13
• Diagonalization: His diagonal argument provided a powerful method for proving that certain sets, such as the real numbers, are uncountable and larger than the set of integers.13

While Cantor established the foundational outlines of set theory, he did not formalize it with axioms. The paradoxes that later emerged from his work were eventually resolved through the development of Zermelo-Fraenkel set theory.24