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The 48 pages that link to Axiom of choice, each with the reason it gives.

• Set theoryRelated: It enables powerful existence proofs and has consequences that cannot be derived from the other usual axioms.

• Zermelo–Fraenkel set theoryRelated: ZF does not include this principle, whose addition changes what the theory can prove.

• Cartesian productRelated: For infinite families, it governs whether a product of nonempty sets must be nonempty.

• Continuum hypothesisRelated: Together with Zermelo–Fraenkel axioms, it forms the standard background for independence results.

• Principia MathematicaRelated: Its treatment illustrates the boundary between logical principles and assumptions needed in mathematics.

• Well-ordering theoremRelated: Choosing one element from each subset yields a well-ordering of any set.

• Ernst ZermeloRelated: Zermelo used this principle to prove that every set can be well-ordered.

• Zorn's lemmaRelated: Zorn's lemma is equivalent to this choice principle in standard set theory.

• Axiom of extensionalityRelated: It operates on families treated as genuine sets, whose identities extensionality fixes by membership.

• Banach–Tarski paradoxRelated: Choosing representatives from rotation orbits produces the nonmeasurable pieces.

• Choice functionRelated: It guarantees choice functions even when no explicit selection rule is given.

• Infinite setRelated: It affects whether certain infinite sets and their subsets can be constructed or well-ordered.

• Tychonoff's theoremRelated: In standard set theory, its full strength is equivalent to Tychonoff’s theorem.

• Axiom of FoundationRelated: Choice is independent of Foundation in standard set theory and addresses a different structural question.

• Gödel's constructible universeRelated: Gödel proved that L satisfies choice, establishing its consistency relative to the base theory.

• Maximal elementRelated: Zorn's lemma, a standard maximal-element principle, is equivalent to the axiom of choice.

• Axiom of constructibilityRelated: The constructible universe satisfies choice, so V = L implies the axiom of choice.

• Axiom of infinityRelated: It is another prominent set-existence principle, but is independent of infinity in standard set theory.

• Felix HausdorffRelated: Its consequences for infinite sets intersected the foundational questions Hausdorff investigated.

• Axiom of Power SetRelated: It is another major Zermelo–Fraenkel extension, logically separate from Power Set.

• Left inverseRelated: For arbitrary sets, extending the forced inverse on f’s image can require choosing outputs elsewhere.

• Nonmeasurable setRelated: A choice function selects the representatives used to construct a Vitali set.

• Existence theoremRelated: It can assert the existence of a selection without specifying how to make each choice.

• Large cardinalRelated: Many standard large-cardinal definitions use choice to characterize cardinal size and structure.

• Aleph numberRelated: It supports the general identification of set cardinalities with aleph numbers.

• Well-orderRelated: In Zermelo–Fraenkel set theory, it is equivalent to the well-ordering theorem.

• Free Will TheoremRelated: The theorem’s free-choice assumption concerns measurement settings, not this set-theoretic axiom.

• Hausdorff paradoxRelated: Choosing one point from each orbit supplies representatives without a geometric selection rule.

• Solovay modelRelated: The model omits full choice, whose consequences include nonmeasurable sets of reals.

• Krull's theoremRelated: In standard set theory, Zorn's lemma—and thus this proof—depends on the axiom of choice.

• Axiom (mathematics and logic)Related: Its independence from other set-theoretic axioms highlights the consequences of adding or withholding an axiom.

• Axiom of limitation of sizeRelated: It is a separate foundational principle and is not what makes proper classes equinumerous with V.