Axiom of choice In mathematics the Axiom of Choice, sometimes called AC, is an axiom used in set theory. The Axiom of Choice says that if you have a group of sets, each containing at least one object, it is possible to take one object out of each of these smaller sets and make a new set, even if there is no rule for selecting objects. You do not always need to use the Axiom of Choice to do this. You do not need to use the Axiom of Choice if the starting set is finite or if the starting set is infinite and has a rule built in for how it can be divided. For example, you could select the smallest number in a group of sets without using the Axiom of Choice even if there are infinite sets because there is a rule for selection. A non mathematical example would be for any (infinite or finite) collection of pairs of shoes, you can identify and pick out the left shoe from each pair. However, for an infinite collection of identical pairs of equal socks, where there is no way to tell apart the socks in each pair, you could not pick from each pair as there is no left or right sock. Therefore there is no rule of selection and so you would need the Axiom of Choice to create a new set. In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle. Errett Bishop, who is notable for developing a framework for constructive analysis, argued that an axiom of choice was constructively acceptable, saying A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence. Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all subsets of real numbers, that are disprovable from the full axiom of choice. Given an ordinal parameter α ≥ ω+2 – for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 – for every set S with Hartogs number less than ωα, 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 axiom of uniformization. Quine's system of axiomatic set theory, New Foundations (NF), takes its name from the title ("New Foundations for Mathematical Logic") of the 1937 article that introduced it. In the NF axiomatic system, the axiom of choice can be disproved. == 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. 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. For proofs, see 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. {\displaystyle (\forall x^{\sigma })(\exists y^{\tau })R(x,y)\to (\exists f^{\sigma \to \tau })(\forall x^{\sigma })R(x,f(x)).} Unlike in set theory, the axiom of choice in type theory is typically stated as an axiom scheme, in which R varies over all formulas or over all formulas of a particular logical form. == Notes == == References == Blass, Andreas (1979). "Injectivity, Projectivity and the Axiom of Choice". Transactions of the American Mathematical Society. 255: 31–59. doi:10.2307/1998165. JSTOR 1998165. Dawson, J. W. (August 2006). "Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödel's Impact on Logic, Mathematics, and Computer Science". doi:10.5486/PMD.1972.19.1-4.37. Halmos, Paul R. (1960). Naive Set Theory. The University Series in Undergraduate Mathematics. Princeton, NJ: van Nostrand Company. Zbl 0087.04403. Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Math. 1876. Berlin: Springer-Verlag. ISBN 978-3-540-30989-5. Howard, Paul; Rubin, Jean E. (1998), Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, Providence, Rhode Island: American Mathematical Society, ISBN 9780821809778 Jech, Thomas (2008) [1973]. The axiom of choice. Mineola, New York: Dover Publications. ISBN 978-0-486-46624-8. Jech, Thomas (1977). "About the Axiom of Choice". In John Barwise (ed.). The Elements of Mathematical Logic. Courier Dover Publications. ISBN 9780486446172. Rubin, Herman; Rubin, Jean E. (April 1970) [1963]. 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