Axiom of choice
Publié le 25/01/2010
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The axiom of choice is a mathematical postulate about sets: for each family of non-empty sets, there exists a function selecting one member from each set in the family. If those sets have no member in common, it postulates that there is a set having exactly one element in common with each set in the family. First formulated in 1904, the axiom of choice was highly controversial among mathematicians and philosophers, because it epitomized 'non-constructive' mathematics. Nevertheless, as time passed, it had an increasingly broad range of consequences in many branches of mathematics.
- Axiome : principe dont on pose la vérité au départ d'une démons-tration chargée d'établir la vérité de ses conséquences logiques.
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choice is consistent with the usual axioms of set theory, but also independent from them.
2 Philosophical questions surrounding the axiom
Today almost all mathematicians accept the axiom of choice, although many still remark when they use it.
Earlier,
Zermelo defended the axiom against its critics on two grounds: its usefulness and its self-evidence.
He remarked
that it had already been widely used, without being formulated explicitly.
Such widespread use, he argued, could
only be explained by the axiom's self-evidence.
Opponents of the axiom had various views.
Some, such as Emile Borel, accepted the axiom when applied to a
denumerable family of sets, but rejected it when applied to a family of larger cardinality, such as the set of all real
numbers.
Others, including René Baire, tentatively accepted the real numbers but argued against the existence of
the set of all subsets of the real numbers, from which subsets Zermelo's choices were made.
Many opponents
rejected the axiom because it provided no rule for making the choices.
Apparently thinking of axioms in the
traditional way as self-evident truths, they often failed to understand that the axiom is needed precisely when no
rule is available.
Opponents were encouraged by the axiom's counter-intuitive consequences.
The most vivid of these was the
Banach-Tarski paradox, whereby a sphere S can be decomposed into a finite number of pieces and reassembled
into two spheres each of the same size as S.
The three-dimensionality is important here, since such a
decomposition cannot be done with a two-dimensional figure such as a circle.
Euclidean geometry also plays a
role, since hyperbolic geometry (in which the parallel postulate is false) allows the various counter-intuitive
decompositions without using the axiom.
The irony for the axiom's opponents was that if set theory is understood in a sufficiently constructivist way, then
the axiom is true.
In the extreme case, when all sets are finite, the axiom is true but trivial.
If set theory is restricted
to Gödel's constructible sets, then again the axiom is true but now distinctly non-trivial.
It is only when set theory
is taken in a non-constructive way that the axiom might be false.
Even in intuitionism, various restricted forms of
the axiom are true, but their meaning is changed since the underlying logic is not first-order logic (see
Intuitionism ).
3 Mathematical theorems that need the axiom
Zermelo introduced the axiom of choice to prove the well-ordering theorem: every set can be well-ordered (see Set
theory ).
Previously Georg Cantor had assumed this well-ordering theorem as a 'law of thought' , but
mathematicians had been sceptical.
Zermelo showed that the well-ordering theorem follows from an axiom that is
conceptually much simpler: the axiom of choice.
The axiom of choice is vital to the arithmetic of infinite cardinal numbers.
In particular, it is equivalent to the
proposition that if two cardinals are unequal, then one is greater than the other.
It is needed in order to define the.
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