Short answers, pulled from the story.
Georg Cantor proved in 1891 that no matter how large a set is, its collection of all possible subsets is always strictly larger. This discovery shattered the intuitive notion that all infinities are equal and revealed an endless hierarchy of magnitudes.
The diagonal argument is a construction where Georg Cantor imagined a table with rows representing numbers and columns representing subsets to create a new set that cannot be found in the original table. This new set serves as the counterexample that destroys any claim of a one-to-one correspondence between a set and its power set.
Cantor's paradox arises when one assumes the existence of a universal set containing all sets, creating an impossible situation where the universal set is both larger than and contained within its own power set. Ernst Zermelo later formalized these ideas in 1908, establishing rules that prevent such contradictions by restricting how sets can be formed.
Bertrand Russell independently developed a similar argument in his 1903 work Principles of Mathematics, though he framed it in terms of propositional functions rather than sets. This became known as Russell's paradox and highlighted the dangers of unrestricted comprehension in set theory.
F. William Lawvere generalized Georg Cantor's theorem to the realm of category theory in the 20th century, creating what is now known as Lawvere's fixed-point theorem. This result applies to any category with finite products and shows that if an endomorphism lacks fixed points, then no object can parameterize all morphisms in that category.