Editing Axiom of choice (section)

==Statements implying the negation of AC==
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We shall abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to validate the negation of some standard ZFC theorems. As any model of ZF¬C is also a model of ZF, it is the case that for each of the following statements, there exists a model of ZF in which that statement is true.

*The negation of the [[Partition principle|weak partition principle]]: There is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all ''known'' models.
*There is a function ''f'' from the real numbers to the real numbers such that ''f'' is not continuous at ''a'', but ''f'' is [[Sequential continuity|sequentially continuous]] at ''a'', i.e., for any sequence {''x<sub>n</sub>''} converging to ''a'', lim<sub>''n''</sub> f(''x<sub>n</sub>'')=f(a).
*There is an infinite set of real numbers without a countably infinite subset.
*The real numbers are a countable union of countable sets.<ref>{{harvnb|Jech|2008|pp=142–144}}, Theorem 10.6 with proof.</ref> This does not imply that the real numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the [[Axiom of countable choice]].
*There is a field with no algebraic closure.
*In all models of ZF¬C there is a vector space with no basis.
*There is a vector space with two bases of different cardinalities.
*There is a free [[complete Boolean algebra]] on countably many generators.<ref name="Stavi, 1974">{{cite journal| first= Jonathan | last=Stavi| year=1974 |title=A model of ZF with an infinite free complete Boolean algebra| journal=[[Israel Journal of Mathematics]]| volume=20| issue= 2| pages=149–163|doi=10.1007/BF02757883|doi-access=| s2cid=119543439}}</ref>
*There is [[Amorphous set|a set that cannot be linearly ordered]].
*There exists a model of ZF¬C in which every set in R<sup>''n''</sup> is [[measurable]].  Thus it is possible to exclude counterintuitive results like the [[Banach–Tarski paradox]] which are provable in ZFC.  Furthermore, this is possible whilst assuming the [[Axiom of dependent choice]], which is weaker than AC but sufficient to develop most of [[real analysis]].
*In all models of ZF¬C, the [[generalized continuum hypothesis]] does not hold.

For proofs, see {{harvtxt|Jech|2008}}.

Additionally, by imposing definability conditions on sets (in the sense of [[descriptive set theory]]) one can often prove restricted versions of the axiom of choice from axioms incompatible with general choice.  This appears, for example, in the [[Moschovakis coding lemma]].