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Axiom of choice - Results requiring ¬AC | | Axiom of choice - Results requiring ¬AC: Encyclopedia II - Axiom of choice - Results requiring ¬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 nu ...
See also:Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes | | | Axiom of choice, Axiom of choice - Independence of AC, Axiom of choice - Quotes, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Results requiring ¬AC, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Weaker forms of AC | | |
| | | Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring ¬AC
Axiom of choice - Results requiring ¬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 for any sequence {xn} converging to a, limn f(xn)=f(a).
- 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.
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.
Other related archives1904, A. K. Dewdney, Algebra, April Fool's Day, Axiom of dependent choice, Baire category theorem, Banach-Alaoglu theorem, Banach–Tarski paradox, Bertrand Russell, Boolean prime ideal theorem, Ernst Zermelo, Functional analysis, General topology, Hahn-Banach theorem, Hausdorff paradox, Hilbert space, Kurt Gödel, Measure theory, Nielsen-Schreier theorem, Paul Cohen, Per Martin-Löf, Scientific American, Set theory, Set-builder notation, Stone's representation theorem for Boolean algebras, Stone-Cech compactification, Trichotomy, Tychonoff space, Tychonoff's theorem, Vitali theorem, Zermelo-Fraenkel set theory, Zorn's lemma, algebraic closure, axiom, axiom of countable choice, axiom of dependent choice, axiom of uniformization, basis, cardinality, category, choice function, choose, closed, closed graph theorem, closure, compact, compactness, complete, constructive logic, constructivists, countably many, determined, field, field extension, functional analysis, game, generalized continuum hypothesis, infinite, infinitely, injection, intuitionistic logic, intuitionistic type theory, law of excluded middle, linear functionals, logically independent, mathematical induction, mathematics, maximal ideal, measurable, metric spaces, model, natural numbers, negation, non-empty, non-measurable sets, nonconstructive, open, open mapping theorem, product, real analysis, real numbers, ring, set, set theory, skeleton, topological spaces, transcendence basis, ultrafilter lemma, uniform spaces, union, vector space, well-ordered, well-ordering principle, well-ordering theorem
 Adapted from the Wikipedia article "Results requiring ¬AC", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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