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The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
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axiom and choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.
In many cases such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin.
For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.
One motivation for this use 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.
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:
Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.
The axiom of choice asserts the existence of such elements ; it is therefore equivalent to:
There are many other equivalent statements of the axiom of choice.
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.
With this alternate notion of choice function, the axiom of choice can be compactly stated as
The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.
However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.
axiom and is
only seldom is it so simple as to be a matter of his obviously parroting some timeworn axiom, common to our culture, which he has evidently heard, over and over, from a parent until he experiences it as part of him.
" In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.
") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.
axiom and avoided
In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation ( Aussonderung ).
axiom and some
The axiom of choice produces these intangibles ( objects that are proven to exist, but which cannot be explicitly constructed ), which may conflict with some philosophical principles.
Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.
One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved.
( The reason for the term " colloquially ", is that the sum or product of a " sequence " of cardinals cannot be defined without some aspect of the axiom of choice.
These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC ( ZF plus AC ).
For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.
Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.
# that some such axiom system is provably consistent through some means such as the epsilon calculus.
Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.
For example, the direct sum of the R < sub > i </ sub > form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.
This is not accomplished by introducing some " new axiom " to quantum mechanics, but on the contrary, by removing the axiom of the collapse of the wave packet.
In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.
The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted.
An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules which expand each symbol into some larger string of symbols, an initial " axiom " string from which to begin construction, and a mechanism for translating the generated strings into geometric structures.
There are many situations where another condition of topological spaces ( such as normality, paracompactness, or local compactness ) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.
The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice ; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.
axiom and varieties
The status of the axiom of choice varies between different varieties of constructive mathematics.
axiom and constructive
The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.
In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).
Thus the axiom of choice is not generally available in constructive set theory.
A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.
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 in constructive set theory.
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
The status of the axiom of choice in constructive mathematics is complicated by the different approaches of different constructivist programs.
One trivial meaning of " constructive ", used informally by mathematicians, is " provable in ZF set theory without the axiom of choice.
However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle ( in the presence of other axioms ), as shown by the Diaconescu-Goodman-Myhill theorem.
Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.
However, the theorem does not rely upon the axiom of choice in the separable case ( see below ): in this case one actually has a constructive proof.
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