Cantor's theorem

Cantor's theorem states that the power set of any set has strictly greater cardinality than the set itself. It applies to finite and infinite sets alike.

Power set: The set of all subsets of a given set. It is the set whose cardinality Cantor’s theorem compares with the original set.
Cardinality: A measure of the size of a set, defined for finite and infinite sets. Cantor’s theorem compares cardinalities rather than ordinary numerical counts.
Uncountable set: An infinite set that cannot be put in bijection with the natural numbers. The power set of the natural numbers is uncountable by Cantor’s theorem.
Georg Cantor: A German mathematician who founded set theory and developed the theory of infinite cardinalities. He proved the result while establishing that infinite sets can have different sizes.
Cantor's diagonal argument: A proof technique that constructs an object differing from every item in a proposed enumeration. Applied to subsets, it produces a subset missing from any proposed list.
Bijection: A function that is both injective and surjective, pairing every element of one set with exactly one of another. Sets have equal cardinality exactly when a bijection exists between them.
Cantor's diagonal argument for real numbers: A diagonal proof that no list can include every infinite sequence of binary digits. It gives a related proof that the real numbers cannot be enumerated.
Set theory: The mathematical study of sets, their elements, and relations between them. Cantor’s theorem became a central result in the study of sets and infinity.
Characteristic function: A function that maps elements in a set to 1 when they belong to a specified subset and 0 otherwise. It represents each subset as a function, enabling the diagonal proof.
Countable set: A set that is finite or can be put in bijection with the natural numbers. The theorem implies that the power set of a countable set is uncountable.

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