 | Axiom of choice: Encyclopedia II - Axiom of choice - Independence of AC
Axiom of choice - Independence of AC
By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF. Consequently, assuming the axiom of choice, or its negation, will never lead to a contradiction that could not be obtained without that assumption.
So the decision whether or not it is appropriate to make use of the axiom of choice in a proof cannot be made by appeal to other axioms of set theory. The decision must be made on other grounds.
One argument given in favor of using the axiom of choice is that it is convenient to use it: using it cannot hurt (cannot result in contradiction) and makes it possible to prove some propositions that otherwise could not be proved.
The axiom of choice is not the only significant statement which is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZF plus the axiom of choice (ZFC). However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF.
One reason that some mathematicians dislike the axiom of choice is that it implies the existence of some bizarre counter-intuitive objects. An example of this is the Banach–Tarski paradox which says in effect that it is possible to "carve up" the 3-dimensional solid unit ball into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell us how to carve up the unit sphere to make this happen, it simply tells us that it can be done.
On the other hand, the negation of the axiom of choice is also bizarre. For example, the statement that for any two sets S and T, the cardinality of S is less than or equal to the cardinality of T or the cardinality of T is less than or equal to the cardinality of S is equivalent to the axiom of choice. Put differently, if the axiom of choice is false, then there are sets S and T of incomparable size: neither can be mapped in a one-to-one fashion onto a subset of the other.
A third possibility is to prove theorems using neither the axiom of choice nor its negation, a tactic preferred in constructive mathematics. Such statements will be true in any model of Zermelo–Fraenkel set theory, regardless of the truth or falsity of the axiom of choice in that particular model. This renders any claim that relies on either the axiom of choice or its negation undecidable. For example, under such an assumption, the Banach–Tarski paradox is neither true nor false: It is impossible to construct a decomposition of the unit ball which can be reassembled into two unit balls, and it is also impossible to prove that it can't be done. However, the Banach–Tarski paradox can be rephrased as a statement about models of ZF by saying, "In any model of ZF in which AC is true, the Banach–Tarski paradox is true." Similarly, all the statements listed below under Results requiring AC are undecidable in ZF, but since each is provable in any model of ZFC, there are models of ZF in which each statement is true.
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
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