Axiom of choice

The axiom of choice is an axiom of Zermelo -Fraenkel set theory. It was first formulated by Ernst Zermelo in 1904. The axiom of choice implies that for each set of non-empty sets there exists a selection function, namely a function of all the non -empty sets an element assigns the same and thus " selects " the. For finite sets, one can conclude that even without this axiom, therefore, the axiom of choice is only interesting for infinite sets.

The axiom of choice

Be a lot of non -empty sets. Then is called a selection function, if the domain has and is:

So choose from each set in exactly one element.

The axiom of choice is then: For any set of non-empty sets there is a choice function.

Example: Let. Then we have. The function defined by

Alternative formulations

• The power set of any set without the empty set has a choice function ( Zermelo 1904).
• Let be a set of pairwise disjoint non -empty sets. Then there are a lot associated with each exactly one common element has ( Zermelo 1907, ZF).

Comments

The axiom of choice postulates the existence of a selection function. But still has no method of how to construct such. One speaks in this case of a weak existence theorem.

For the following cases a selection function also exists without the axiom of choice:

• For a finite set of non -empty sets, it is trivial to specify a selection function: select from lots of any particular element of what is possible. One does not need the axiom of choice for this purpose. A formal proof would use induction on the size of the finite set.
• For quantities of non-empty subsets of the natural numbers, it is also easily possible: select from any subset of the smallest element. Similarly, one can specify for a set of closed subsets of the real numbers an explicit selection function (without using the axiom of choice ), by choosing the (possibly positive ) element with the smallest absolute value of approximately from each set.
• Even for sets of intervals of real numbers is a selection function defined: One chooses from each interval from the center.

In which cases the axiom of choice is relevant, may be illustrated by the following examples:

• One can already for a general countable set of two quantities in ZF (not ZFC, that is, without the axiom of choice ) does not prove the existence of a selection function.
• The same is true about the existence of a selection function for the set of all nonempty subsets of real numbers.

However, there are weakening the axiom of choice, which does not imply this, but show for cases such as the two examples the existence of, for example for the first case the axiom CC ( for " Countable Choice " ), which states that a choice function exists if the amount of family is countable.

Kurt Gödel showed in 1938 that the axiom of choice in the context of Zermelo -Fraenkel set theory gives no contradiction if one assumes the consistency of all the other axioms. 1963 but showed Paul Cohen, that the negation (ie the "opposite " ) will not cause the axiom of choice to a contradiction. Both assumptions are therefore from the formalistic point of view, acceptable.

The axiom of choice is accepted by the vast majority of mathematicians. In many branches of mathematics, including newer ones like the non-standard calculus, it leads to particularly aesthetic results. However, the Constructivist Mathematics is a mathematics branch, which deliberately avoids the axiom of choice. In addition, there are other mathematicians, including many of the theoretical physics related parties that do not use the axiom of choice also, especially because of contrasting intuitive consequences such as the Banach - Tarski paradox. This leads to the question of whether sentences, usually for their proof the axiom of choice is used, such as the Hahn- Banach let soften so that they can be proved without the axiom of choice, yet cover all major applications.

For axiom of choice equivalent sentences

Substituting the ZF axioms ahead, then there are a number of important theorems that are equivalent to the axiom of choice. The most important among them are the Lemma of Zorn and the well-ordering theorem. Zermelo introduced the axiom of choice to formalize the proof of the well-ordering theorem. The names " lemma " and " set " stem from the fact that these formulations do not appear immediately as insightful as the axiom of choice itself

• Set theory Well-ordering theorem: Every set can be well ordered.
• If an infinite set, then and have the same cardinality.
• Trichotomy: Two sets have the same cardinality, or either one of the two quantities has a smaller cardinality than the other.
• The Cartesian product of a non-empty family of non -empty sets is non-empty.
• König's theorem: Simplified: The sum of a sequence of cardinal numbers is strictly smaller than the product of a series of larger cardinal numbers.
• Every surjective function has a right inverse.
• Lemma of Teichmüller - Tukey: If M is a non-empty set of finite character, so there is regarding the set inclusion a maximal element.

• Lemma of Zorn: Every non-empty partially ordered set in which every chain ( ie every totally ordered subset ) has an upper bound contains at least one maximal element.
• Hausdorff's maximum chain sentence: In an ordered set, each chain can be extended to a maximum chain.
• Hausdorff's maximum chain kit ( attenuated): In a minor amount of at least one maximal chain exists.

• Each generating set of a vector space contains a basis of.
• Every vector space has a basis.
• Each ring with unit element, which is not the zero ring has a maximal ideal.

• Each (infinite ) undirected, connected graph has a spanning tree.

• Set of Tychonoff: The product of compact spaces is itself compact - but only if it does not require the Hausdorf fig shaft for compactness.
• In the product topology: The completion of a product of subsets is equal to the product of the financial statements of the subsets.
• The product of complete uniform spaces is complete.