View source for Axiom of choice

{{MSC|03E25|03E35}}
{{TEX|done}}


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 lemma|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) [[Tikhonov theorem|Tikhonov's theorem]]: 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====
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|valign="top"|{{Ref|Je}}||valign="top"| T.J. Jech, "Lectures in set theory: with particular emphasis on the method of forcing", ''Lect. notes in math.'', '''217''', Springer (1971) {{MR|0321738}} {{ZBL|0236.02048}}
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|valign="top"|{{Ref|Je2}}||valign="top"| T.J. Jech, "The axiom of choice", North-Holland (1973) {{MR|0396271}} {{ZBL|0259.02051}}
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|valign="top"|{{Ref|Je3}}||valign="top"| T.J. Jech, "Set theory", Acad. Press (1978) (Translated from German) {{MR|0506523}} {{ZBL|0419.03028}}
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|valign="top"|{{Ref|FrBaLe}}||valign="top"| A.A. Fraenkel, Y. Bar-Hillel, A. Levy, "Foundations of set theory", North-Holland (1973) {{MR|0345816}} {{ZBL|0248.02071}}
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