Statements Consistent With The Negation of AC
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will 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 prove the negation of some standard facts. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.
- There exists a model of ZF¬C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i.e., for any sequence {xn} converging to a, limn f(xn)=f(a).
- There exists a model of ZF¬C which has an infinite set of real numbers without a countably infinite subset.
- There exists a model of ZF¬C in which real numbers are a countable union of countable sets.
- There exists a model of ZF¬C in which there is a field with no algebraic closure.
- In all models of ZF¬C there is a vector space with no basis.
- There exists a model of ZF¬C in which there is a vector space with two bases of different cardinalities.
- There exists a model of ZF¬C in which there is a free complete boolean algebra on countably many generators.
For proofs, see Thomas Jech, The Axiom of Choice, American Elsevier Pub. Co., New York, 1973.
- There exists a model of ZF¬C in which every set in Rn 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.
Read more about this topic: Axiom Of Choice
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