axiom of choice

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The axiom of choice is one of the statements of set theory, postulating the existence of a choice function on any family of nonempty sets. That is, for any collection (possibly infinite) of pairwise disjoint nonempty sets, there exists a set containing exactly one element from each set in the collection. For a finite number of sets it seems trivial, but for infinite ones it gives a non-constructive result: we know that a choice is possible, but we cannot describe it. It is equivalent to Zermelo's well-ordering principle and Zorn's lemma.

History

First explicitly formulated by the German mathematician Ernst Zermelo in 1904 to prove the well-ordering theorem. This sparked heated debates because the axiom seemed unnatural. Over time, most mathematicians accepted it, although some areas of mathematics are built without it.

How it works

Suppose we have a set of nonempty sets (a bag of bags). The axiom of choice guarantees the existence of a new set—a 'mixed platter' into which we put exactly one object from each bag. Mathematicians can calmly say: 'Consider a choice function...' and proceed further, without bothering with specific rules.

💡 Using the axiom of choice, one can mathematically 'double' a ball: cut it into finitely many pieces, move them, and obtain two balls of the same radius. This Banach–Tarski paradox shows how powerful and counterintuitive the axiom is.

The axiom of choice (AC) in formal notation: for any family X of nonempty sets there exists a function f: X → ⋃X such that for every A ∈ X, f(A) ∈ A. In other words: every family of nonempty sets has a choice function. Within Zermelo–Fraenkel set theory (ZF), AC is independent of the other axioms: one can add either it or its negation without obtaining a contradiction (results of Gödel and Cohen). AC is equivalent to the well-ordering principle, Zorn's lemma, the maximal ideal theorem, and many other statements. In mathematical practice, AC is used everywhere, but it gives rise to non-constructive existence proofs and paradoxes, for example the existence of non-measurable sets (Vitali) and the Banach–Tarski paradox.

History

First explicitly formulated by Ernst Zermelo in 1904 to prove that every set can be well-ordered. This claim was immediately challenged by Émile Borel, Henri Lebesgue and other leading analysts. In 1938, Kurt Gödel showed that AC is consistent with ZF, and in 1963 Paul Cohen proved that the negation of AC is also compatible with ZF. Thus, the axiom of choice is independent. Today, the overwhelming majority of mathematicians use AC, although alternative approaches (constructivism, topos theory) exist where it is avoided.

How it works

Functionally, AC allows non-constructive constructions. In proving theorems, one often says: 'For each i, choose an element x_i from the set A_i' without specifying a choice rule. This is valid if the axiom of choice is accepted. Without it, such reasoning is inadmissible. A typical example: the proof of the existence of a Hamel basis in an arbitrary vector space—we successively choose linearly independent vectors, and at infinite steps AC is required. Technically, the axiom does not indicate how to find the choice; it simply postulates its existence.

\forall X (\emptyset \notin X \implies \exists f: X \to \bigcup X \text{ such that } \forall A \in X, f(A) \in A)

X is a family of nonempty sets, ⋃X is the union of all elements of X, f is a choice function assigning to each A ∈ X an element from A.

Open problems & fun facts

Independence from ZF: AC is neither provable nor disprovable in ZF.; Paradoxical consequences (Banach–Tarski) raise philosophical questions about the nature of mathematical existence.; In constructivist mathematics, AC is rejected, which requires rebuilding many areas.

💡 The statement 'the Tychonoff product of compact spaces is compact' is equivalent to the axiom of choice, although the connection is far from obvious. Likewise, 'every ring with a unit contains a maximal ideal' is also equivalent to AC.

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axiom of choice

History

How it works

History

How it works

History

How it works

Open problems & fun facts

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