Axiom of choice
Introduction
The **Axiom of Choice** is a fundamental principle in set theory, a branch of mathematical logic that studies sets, which are collections of objects. This axiom is pivotal in many areas of mathematics, particularly in analysis, topology, and algebra. It states that given a collection of non-empty sets, it is possible to select exactly one element from each set, even if there is no explicit rule for making the selection. The axiom is controversial because it asserts the existence of a selection function without necessarily providing a constructive method for finding it.
Historical Context
The Axiom of Choice was formulated in the early 20th century by the German mathematician Ernst Zermelo, who introduced it in 1904 to prove the well-ordering theorem. This theorem states that every set can be well-ordered, meaning its elements can be arranged in a sequence where every subset has a least element. The axiom was initially met with skepticism because it allowed for the existence of sets that could not be explicitly constructed. Despite this, it has become an accepted part of modern set theory and is included in the standard Zermelo-Fraenkel set theory (ZF), often denoted as ZFC when combined with the Axiom of Choice.
Formal Definition
In formal terms, the Axiom of Choice can be stated as follows: For any set \( X \) of non-empty sets, there exists a choice function \( f \) defined on \( X \) such that for every set \( A \) in \( X \), \( f(A) \) is an element of \( A \). This can be expressed symbolically as:
\[ \forall X \left( \forall A \in X, A \neq \emptyset \implies \exists f: X \to \bigcup X, \forall A \in X, f(A) \in A \right) \]
Equivalents and Consequences
The Axiom of Choice is equivalent to several other mathematical propositions, which are often used interchangeably in proofs. Some of these include:
- **Zorn's Lemma**: If every chain (totally ordered subset) in a partially ordered set has an upper bound, then the set contains at least one maximal element.
- **Well-Ordering Theorem**: Every set can be well-ordered.
- **Tychonoff's Theorem**: The product of any collection of compact topological spaces is compact.
Each of these statements, while seemingly different, can be proven to imply the others when the Axiom of Choice is assumed.
Applications in Mathematics
The Axiom of Choice is indispensable in various fields of mathematics:
Analysis
In analysis, the Axiom of Choice is used to prove the existence of bases for vector spaces, even infinite-dimensional ones. It is also crucial in the proof of the Hahn-Banach theorem, which extends linear functionals.
Topology
In topology, the Axiom of Choice is used in the proof of Tychonoff's Theorem, which states that the product of any collection of compact spaces is compact. This theorem is a cornerstone of general topology and has far-reaching implications in functional analysis and other areas.
Algebra
In algebra, the Axiom of Choice is used to show that every vector space has a basis, a result that is not constructively provable without the axiom. It also plays a role in the proof of the existence of algebraic closures of fields.
Controversies and Criticism
The Axiom of Choice has been a subject of controversy since its inception. Critics argue that it allows for the existence of non-constructive objects, which cannot be explicitly described or constructed. This has led to debates about its validity and applicability, particularly in constructive mathematics, which avoids non-constructive proofs.
One of the most famous paradoxes arising from the Axiom of Choice is the Banach-Tarski Paradox, which states that a solid ball in three-dimensional space can be divided into a finite number of non-overlapping pieces, which can then be rearranged to form two identical copies of the original ball. This counterintuitive result challenges our understanding of volume and measure.
Alternatives and Variants
Several alternatives to the Axiom of Choice have been proposed, each with its own implications and limitations:
- **Axiom of Determinacy**: An alternative to the Axiom of Choice, which is incompatible with it, focusing on games of infinite length.
- **Dependent Choice**: A weaker form of the Axiom of Choice, sufficient for many purposes in analysis.
- **Axiom of Countable Choice**: A restricted version of the Axiom of Choice, applicable only to countable collections of sets.
These alternatives are often studied in the context of different set-theoretical frameworks, such as constructive set theory and intuitionistic logic.