The power set is always strictly larger than the set. Proof by diagonal: for f: X → P(X), the set D = {x : x ∉ f(x)} is not in the image of f. Yields the infinite hierarchy of cardinalities |X| < |P(X)| < |P(P(X))| < ⋯ and motivates the…
The power set is always strictly larger than the set. Proof by diagonal: for f: X → P(X), the set D = {x : x ∉ f(x)} is not in the image of f. Yields the infinite hierarchy of cardinalities |X| < |P(X)| < |P(P(X))| < ⋯ and motivates the…