Editing Axiom of choice (section)

==Weaker forms==
There are several weaker statements that are not equivalent to the axiom of choice but are closely related. One example is the [[axiom of dependent choice]] (DC). A still weaker example is the [[axiom of countable choice]] (AC<sub>ω</sub> or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary [[mathematical analysis]], and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

Given an ordinal parameter &alpha; ≥ &omega;+2 &mdash; for every set ''S'' with rank less than &alpha;, ''S'' is well-orderable. Given an ordinal parameter &alpha; ≥ 1 &mdash; for every set ''S'' with [[Hartogs number]] less than &omega;<sub>&alpha;</sub>, ''S'' is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely.

Other choice axioms weaker than axiom of choice include the [[Boolean prime ideal theorem]] and the [[Uniformization (set theory)|axiom of uniformization]].  The former is equivalent in ZF to [[Alfred Tarski|Tarski]]'s 1930 [[ultrafilter lemma]]: every [[Filter (set theory)|filter]] is a subset of some [[Ultrafilter (set theory)|ultrafilter]].

===Results requiring AC (or weaker forms) but weaker than it===<!-- This section is linked from [[Basis (linear algebra)]] -->
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics where it shows up. Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC). Equivalently, these statements are true in all models of ZFC but false in some models of ZF.

*[[Set theory]]
**The [[ultrafilter lemma]] (with ZF) can be used to prove the Axiom of choice for finite sets: Given <math>I \neq \varnothing</math> and a collection <math>\left(X_i\right)_{i \in I}</math> of non-empty {{em|finite}} sets, their product <math>\prod_{i \in I} X_{i}</math> is not empty.<ref name="Muger2020">{{cite book|last=Muger|first= Michael|title=Topology for the Working Mathematician|year=2020}}</ref>
**The [[union (set theory)|union]] of any countable family of [[countable sets]] is countable (this requires [[Axiom of countable choice|countable choice]] but not the full axiom of choice).
**If the set ''A'' is [[infinite set|infinite]], then there exists an [[injective function|injection]] from the [[natural number]]s '''N''' to ''A'' (see [[Dedekind infinite]]).<ref>It is shown by {{harvnb|Jech|2008|pp=119–131}}, that the axiom of countable choice implies the equivalence of infinite and Dedekind-infinite sets, but that the equivalence of infinite and Dedekind-infinite sets does not imply the axiom of countable choice in ZF.</ref>
**Eight definitions of a [[finite set#Other concepts of finiteness|finite set]] are equivalent.<ref>It was shown by {{harvnb|Lévy|1958}} and others using Mostowski models that eight definitions of a finite set are independent in ZF without AC, although they are equivalent when AC is assumed. The definitions are I-finite, Ia-finite, II-finite, III-finite, IV-finite, V-finite, VI-finite and VII-finite. I-finiteness is the same as normal finiteness. IV-finiteness is the same as Dedekind-finiteness.</ref>
**Every infinite [[determinacy#Basic notions|game]] <math>G_S</math> in which <math>S</math> is a [[Borel set|Borel]] subset of [[Baire space (set theory)|Baire space]] is [[determinacy#Basic notions|determined]].
* Every infinite [[cardinal number|cardinal]] ''κ'' satisfies 2×''κ'' = ''κ''.<ref>{{cite journal|last=Sageev|first=Gershon|title=An independence result concerning the axiom of choice|journal=Annals of Mathematical Logic|volume=8|issue=1–2|date=March 1975|pages=1–184|doi=10.1016/0003-4843(75)90002-9}}</ref>
*[[Measure theory]]
**The [[Vitali set|Vitali theorem]] on the existence of [[non-measurable set]]s, which states that there exists a subset of the [[real numbers]] that is not [[Lebesgue measurable]].
**There exist Lebesgue-measurable subsets of the real numbers that are not [[Borel set]]s. That is, the Borel σ-algebra on the real numbers (which is generated by all real intervals) is strictly included the Lebesgue-measure σ-algebra on the real numbers.
**The [[Hausdorff paradox]].
**The [[Banach–Tarski paradox]].
*[[Algebra]]
**Every [[field (mathematics)|field]] has an [[algebraic closure]].
**Every [[field extension]] has a [[transcendence basis]].
**Every infinite-dimensional [[vector space]] contains an infinite linearly independent subset (this requires [[Axiom of dependent choice|dependent choice]], but not the full axiom of choice).
**[[Stone's representation theorem for Boolean algebras]] needs the [[Boolean prime ideal theorem]].
**The [[Nielsen–Schreier theorem]], that every subgroup of a free group is free.
**The additive groups of '''[[real numbers|R]]''' and '''[[complex number|C]]''' are isomorphic.<ref>{{cite web|url=http://www.cs.nyu.edu/pipermail/fom/2006-February/009959.html|title=[FOM] Are (C,+) and (R,+) isomorphic|date=21 February 2006 }}</ref><ref>{{cite journal|title=A consequence of the axiom of choice|first=C. J.|last=Ash|journal=Journal of the Australian Mathematical Society|year=1975 |volume=19 |issue=3 |pages=306–308 |doi=10.1017/S1446788700031505 |s2cid=122334025 |doi-access=free}}</ref>
*[[Functional analysis]]
**The [[Hahn–Banach theorem]] in [[functional analysis]], allowing the extension of [[linear map|linear functionals]].
**The theorem that every [[Hilbert space]] has an orthonormal basis.
**The [[Banach–Alaoglu theorem]] about [[compactness]] of sets of functionals.
**The [[Baire category theorem]] about [[complete space|complete]] [[metric space]]s, and its consequences, such as the [[open mapping theorem (functional analysis)|open mapping theorem]] and the [[closed graph theorem]].
**On every infinite-dimensional topological vector space there is a [[discontinuous linear map]].
*[[General topology]]
**A uniform space is compact if and only if it is complete and totally bounded.
**Every [[Tychonoff space]] has a [[Stone–Čech compactification]].
*[[Mathematical logic]]
**[[Gödel's completeness theorem]] for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of first-order sentences can be extended to a maximal consistent set.
**The [[compactness theorem]]: If <math>\Sigma</math> is a set of [[First-order predicate calculus|first-order]] (or alternatively, [[Propositional calculus|zero-order]]) [[Sentence (mathematical logic)|sentences]] such that every [[Finite set|finite]] subset of <math>\Sigma</math> has a [[Model (model theory)|model]], then <math>\Sigma</math> has a model.{{sfn|Schechter|1996|pp=391-392}}

===Possibly equivalent implications of AC===
There are several historically important set-theoretic statements implied by AC whose equivalence to AC is open. Zermelo cited the partition principle, which was formulated before AC itself, as a justification for believing AC. In 1906, Russell declared PP to be equivalent, but whether the partition principle implies AC is the oldest open problem in set theory,<ref>{{cite web | url=https://karagila.org/2014/on-the-partition-principle/ | title=On the Partition Principle }}</ref> and the equivalences of the other statements are similarly hard old open problems. In every ''known'' model of ZF where choice fails, these statements fail too, but it is unknown whether they can hold without choice.

*[[Set theory]]
**Partition principle: if there is a [[Surjective function|surjection]] from ''A'' to ''B'', there is an [[Injective function|injection]] from ''B'' to ''A''. Equivalently, every [[Partition of a set|partition]] ''P'' of a set ''S'' is less than or equal to ''S'' in size.
**Converse [[Schröder–Bernstein theorem]]: if two sets have surjections to each other, they are equinumerous.
**Weak partition principle: if there is an [[Injective function|injection]] and a [[Surjective function|surjection]] from ''A'' to ''B'', then ''A'' and ''B'' are equinumerous. Equivalently, a partition of a set ''S'' cannot be strictly larger than ''S''. If WPP holds, this already implies the existence of a non-measurable set. Each of the previous three statements is implied by the preceding one, but it is unknown if any of these implications can be reversed.
**There is no infinite decreasing sequence of cardinals. The equivalence was conjectured by Schoenflies in 1905.
*[[Abstract algebra]]
**[[Hahn embedding theorem]]: Every ordered abelian group ''G'' order-embeds as a subgroup of the additive group <math>\mathbb{R}^\Omega</math> endowed with a [[lexicographical order]], where Ω is the set of Archimedean equivalence classes of ''G''. This equivalence was conjectured by Hahn in 1907.