Short answers, pulled from the story.
Georg Cantor established that for any set, the collection of all its subsets must be strictly larger than the original set itself. A set with n elements generates exactly 2^n subsets when including the empty set. The inequality 2^n exceeds n for every non-negative integer value.
Cantor constructed a specific subset known as the diagonal set to disprove surjection. He defined this set D such that an element x belongs to D if and only if x does not belong to f(x). This contradiction proves no function can map onto every possible subset.
Georg Cantor published his proof in a paper titled Über eine elementare Frage der Mannigfaltigkeitslehre during 1891. The article appeared alongside his earlier work on the uncountability of real numbers. Lawrence Paulson noted in 1992 that automated theorem provers struggled to discover this specific diagonal construction without significant human direction.
Cantor's paradox arises when assuming a universal set containing all sets exists because its power set would be larger yet contained within it by definition. This creates a logical contradiction where the whole is smaller than one of its parts. Russell's paradox derives from instantiating the function with the identity mapping.
Lawvere developed a fixed-point theorem generalizing Cantor's result to any category with finite products. His abstract framework extends the diagonal argument beyond sets into broader mathematical structures. The theorem applies whenever a morphism fails to have a fixed point.