Axiom of choice

2020 Mathematics Subject Classification: Primary: 03E25 Secondary: 03E35 [MSN][ZBL]

One of the axioms in set theory. It states that for any family $F$ of non-empty sets there exists a function $f$ such that, for any set $S$ from $F$, one has $f(S)\in S$ ($f$ is called a choice function on $F$). For finite families $F$ the axiom of choice can be deduced from the other axioms of set theory (e.g. in the system ZF).

The axiom of choice was explicitly formulated by E. Zermelo (1904) and was objected to by many mathematicians. This is explained, first, by its purely existential character which makes it different from the remaining axioms of set theory and, secondly, by some of its implications which are "unacceptable" or even contradict intuitive "common sense" . Thus, the axiom of choice implies the existence of a Lebesgue non-measurable set of real numbers; the Banach–Tarski paradox: the existence of three subdivisions of the sphere $B$ $$B=U_1\cup\dots\cup U_n,$$

$$B=V_1\cup\dots\cup V_m,$$

$$B=X_1\cup\dots\cup X_{n+m},$$ such that $U_i$ is congruent with $X_i$, $1\le i \le n$, and $V_j$ is congruent with $X_{n+j}$, $1\le j\le m$. Thus, the sphere $B$ is divisible into a finite number of parts $X_1,\dots,X_{n+m}$ which can be moved in space to form two spheres identical to it.

Many postulates equivalent to the axiom of choice were subsequently discovered. Among these are: 1) The well-ordering theorem: On any set $X$ there exists a total order $R\subseteq X\times X$ such that any non-empty set $U\subset X$ contains a least element in the sense of the relation $R$; 2) the maximality principle (Zorn's lemma): If any totally ordered subset $U$ of a partially ordered set $X$ is bounded from above, $X$ contains a maximal element; 3) any non-trivial lattice with a unit element has a maximal ideal; 4) the product of compact topological spaces is compact; and 5) any infinite set $X$ has the same cardinality as $X\times X$.

The axiom of choice does not contradict the other axioms of set theory (e.g. the system ZF) and cannot be logically deduced from them if they are non-contradictory. The axiom of choice is extensively employed in classical mathematics. Thus, it is used in the following theorems. 1) Each subgroup of a free group is free; 2) the algebraic closure of an algebraic field exists and is unique up to an isomorphism; and 3) each vector space has a basis. It is also used in: 4) the equivalence of the two definitions of continuity of a function at a point (the $\epsilon - \delta$-definition and the definition by limits of sequences) and in proving 5) the countable additivity of the Lebesgue measure. The last two theorems follow from the countable axiom of choice (the formulation of the axiom includes the condition of countability of the family $F$). It was proved that the theorems 1) to 5) are not deducible in the system ZF if ZF is non-contradictory.

A model of set theory has been constructed which meets the countable axiom of choice and in which each set of numbers is Lebesgue-measurable. This model was constructed on the assumption that the system ZF does not contradict the axiom of the existence of an inaccessible cardinal number.

References

[Je]	T.J. Jech, "Lectures in set theory: with particular emphasis on the method of forcing", Lect. notes in math., 217, Springer (1971) MR0321738 Zbl 0236.02048
[Je2]	T.J. Jech, "The axiom of choice", North-Holland (1973) MR0396271 Zbl 0259.02051
[Je3]	T.J. Jech, "Set theory", Acad. Press (1978) (Translated from German) MR0506523 Zbl 0419.03028
[FrBaLe]	A.A. Fraenkel, Y. Bar-Hillel, A. Levy, "Foundations of set theory", North-Holland (1973) MR0345816 Zbl 0248.02071