Axiom of Choice
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 non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every set
I
{\displaystyle I}
and every
I
{\displaystyle I}
-indexed family
(
S
i
)
i
∈
I
{\displaystyle (S_{i})_{i\in I}}
of nonempty sets, there exists an
I
{\displaystyle I}
-indexed set
(
x
i
)
i
∈
I
{\displaystyle (x_{i})_{i\in I}}
of elements of
∪
i
∈
I
S
i
{\displaystyle \cup _{i\in I}S_{i}}
such that
x
i
∈
S
i
{\displaystyle x_{i}\in S_{i}}
for every
i
∈
I
{\displaystyle i\in I}
. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
The axiom of choice is equivalent to the statement that every partition has a transversal.
In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite (in which induction can be applied), or if a canonical rule on how to choose the elements is available – some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets are collected from the natural numbers, it will always be possible to form a choice function from choosing the smallest element from each set to produce a set; the axiom of choice is not needed here. On the other hand, for the collection of all non-empty subsets of the real numbers, there is no known canonical rule by which one can choose one element from each of these subsets. In that case, the axiom of choice must be invoked to construct the desired choice function.
Bertrand Russell coined an analogy: for any (even infinite) collection of unordered pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function without using the axiom of choice. However, for an infinite collection of unordered pairs of socks (assumed to have no distinguishing features such as being a left sock rather than a right sock), there is no natural (i.e., canonical) way of choosing one sock from each pair, so one must appeal to the axiom of choice to construct the desired choice function.
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). One motivation for this is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. While some varieties of constructive mathematics avoid the axiom of choice, others embrace it.
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