In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Tarski's theorem, proved by , states that in
ZF the theorem "For every infinite set
, there is a
bijective map
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...
between the sets
and
" implies the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
. The opposite direction was already known, thus the theorem and axiom of choice are equivalent.
Tarski told that when he tried to publish the theorem in ''
Comptes Rendus de l'Académie des Sciences de Paris'',
Fréchet and
Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest.
Proof
The goal is to prove that the axiom of choice is implied by the statement "for every infinite set
".
It is known that the
well-ordering theorem
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the order ...
is equivalent to the axiom of choice; thus it is enough to show that the statement implies that for every set
there exists a
well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then calle ...
.
Since the collection of all
ordinals such that there exists a
surjective function
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a ...
from
to the ordinal is a set, there exists an infinite ordinal,
such that there is no
surjective function
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a ...
from
to
We assume
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
that the sets
and
are
disjoint.
By the initial assumption,
thus there exists a
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
For every
it is impossible that
because otherwise we could define a surjective function from
to
Therefore, there exists at least one ordinal
such that
so the set
is not empty.
We can define a new function:
This function is well defined since
is a non-empty set of ordinals, and so has a minimum.
For every
the sets
and
are disjoint.
Therefore, we can define a well order on
for every
we define
since the image of
that is,