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In mathematics, the axiom of choice is an axiom in set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the time. It states the following:
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Stated more formally:
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Another formulation of the axiom of choice (AC) states:
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For many years, the axiom of choice was used implicitly. For example, a proof could use a set S that was previously demonstrated to contain only non-empty sets and claim "because X is non-empty for all X in S, let F(X) be one of the members of X for all X in S." Here, the use of F requires the axiom of choice.
The principle seems obvious: if you have several boxes lying around with at least one item in each box, the axiom simply states that you can choose one item out of each of them. Although the statement sounds straightforward there's a controversy over what it means to choose something from these sets. As an example, let us look at some sample sets.
And therein lies the crux of the axiom. All it states is that there is some function f that can choose an element out of each set in the collection. It gives you no indication about how the function would be defined, it simply mandates its existence. Theorems whose proofs involve the axiom of choice are always nonconstructive: they postulate the existence of something without telling you how to get it.
The axiom of choice has been proven to be logically independent of the remaining axioms of set theory; that is, it can be neither proven nor disproven from them (unless those remaining axioms contain an unknown contradiction). This is the result of work by Kurt Gödel and Paul Cohen. There are thus no contradictions if you choose not to accept the axiom of choice; however, most mathematicians accept either it, or a weakened variant of it, because it makes their jobs easier. Despite this, there is some study of systems in which the axiom of choice is either not true or at least not assumed (see also axiom of regularity). In these cases it is important to be aware which proofs in mathematics use the axiom of choice and which do not.
One of the reasons that some mathematicians do not particularly like 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 amounts to saying 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 you how to carve up the unit sphere to make this happen, it simply tells you that it can be done.
One of the most interesting aspects of the axiom of choice is the sheer number of places in mathematics that it shows up. There are also a remarkable number of statements that are equivalent to the axiom of choice, most important among them Zorn's lemmaZorn's lemma also known as the Kuratowski-Zorn lemma is a theorem of set theory that states: Every partially ordered set in which every chain (i. totally ordered subset) has an upper bound contains at least one maximal element. It is named after the mathe and the well-ordering theoremSet theory The well-ordering theorem (not to be confused with the well-ordering axiom) states that every set can be well-ordered. This is important because it makes every set susceptible to the powerful technique of transfinite induction. Georg Cantor con: every set can be well-ordered, and in fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle.
Several central theorems in various branches of mathematics require the axiom of choice (or one of its weaker versions, such as the Boolean prime ideal theoremIn mathematics, a number of so called prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. Among the most popular statements of this form is the Boolean prime ideal theorem which states that ideals i, the axiom of countable choiceSet theory The axiom of countable choice or axiom of denumerable choice is an axiom of set theory similar to the axiom of choice. It states that a countable collection of sets must have a choice function. Paul Cohen showed that this is not provable in ZF., or the axiom of dependent choiceSet theory In mathematics, the axiom of dependent choice is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X the). These branches are
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