In mathematics, the axiom of choice, or AC, is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originally controversial, it is now used without embarrassment by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, that either reject the axiom of choice, or even investigate consequences of axioms inconsistent with AC. Intuitively speaking, AC says that given a collection of bins, each containing at least one object, then exactly one ob ...

Including:

• Axiom of choice - Statement
• Axiom of choice - Usage
• Axiom of choice - Independence of AC
• Axiom of choice - Weaker forms of AC
• Axiom of choice - Results requiring AC or weaker forms
• Axiom of choice - Results requiring ¬AC
• Axiom of choice - Results requiring choice in intuitionistic logic though not classically
• Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia - Axiom of choice

Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic, though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring choice in intuitionistic logic, though not classically

There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will 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 prove the negation of some standard facts. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true. There exists a model of ZF¬C in which there is a function f from the real numbers to the real nu ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring ¬AC

Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X." In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo. Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Usage

Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring choice in intuitionistic logic though not classically

The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X. Stated more formally: Let X be a set of non-empty sets. Then there exists a choice function f defined on X. In other words, there exists a function f defined on X, such that for each set s in X, f(s) is an element of s. Another formulation of the axiom of choice states: Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one eleme ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Statement

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 s ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Independence of AC