Equivalence of Formulations of Axiom of Choice

Theorem

The following formulations of the Axiom of Choice are equivalent:

Formulation 1

For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set.

$\ds \forall s: \paren {\O \notin s \implies \exists \paren {f: s \to \bigcup s}: \forall t \in s: \map f t \in t}$

That is, one can always create a choice function for selecting one element from each element of the set.

Formulation 2

Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

Then there exists an indexed family $\family {x_i}_{i \mathop \in I}$ such that:

$\forall i \in I: x_i \in X_i$

That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.

Formulation 3

Let $\SS$ be a set of non-empty pairwise disjoint sets.

Then there exists a set $C$ such that for all $S \in \SS$, $C \cap S$ has exactly one element.

Symbolically:

$\forall s: \paren {\paren {\O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O} \implies \exists c: \forall t \in s: \exists x: t \cap c = \set x}$

Formulation 4

Let $A$ be a non-empty set.

Then there exists a mapping $f: \powerset A \to A$ such that:

for every non-empty proper subset $x$ of $A$: $\map f x \in x$

where $\powerset A$ denotes the power set of $A$.

Proof

Formulation 1 implies Formulation 2

Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

By hypothesis, Formulation 1 holds.

That is, there exists a choice function on every set of non-empty sets.

Let $f$ be a choice function on $\set{X_i}$.

Let $x_i = \map f {X_i}$.

By definition of choice function, each $x_i \in X_i$.

Therefore, $\family {x_i}_{i \mathop \in I}$ satisfies Formulation 2.

$\Box$

Formulation 2 implies Formulation 1

Suppose that Formulation 2 holds.

That is, the Cartesian product of a non-empty family of non-empty sets is non-empty.

Let $\CC$ be a non-empty set of non-empty sets.

$\CC$ may be converted into an indexed set by using $\CC$ itself as the indexing set and using the identity mapping on $\CC$ to do the indexing.

Then the Cartesian product of all the sets in $\CC$ has at least one element.

An element of such a Cartesian product is a mapping, that is a family whose domain is the indexing set which in this context is $\CC$.

The value of this mapping at each index is an element of the set which bears that index.

So we have a mapping $f$ whose domain is $\CC$ such that:

$A \in \CC \implies \map f A \in A$

Now let $\CC$ be the set of all non-empty subsets of some set $X$.

Then the assertion means that there exists a mapping $f$ whose domain is $\powerset X \setminus \set \O$ such that:

$A \in \powerset X \setminus \set \O \implies \map f A \in A$

That is, Formulation 1 holds.

$\Box$

Formulation 1 implies Formulation 3

Let $\SS$ be the set:

$\SS = \set {s: \O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O}$

Let $c$ be a choice function on $\SS$ and consider the image set $c \sqbrk \SS$:

$c \sqbrk \SS = \set {\map c s: \O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O}$

By the definition of choice function:

$\map c s \in s$

By construction of $\SS$, for any $s \in \SS$:

$s \cap c \sqbrk \SS = \set {\map c s}$

$\Box$

Formulation 3 implies Formulation 1

Let $\BB$ be a non-empty indexed family of non-empty sets indexed by $\II$.

Consider sets of the following form:

$\CC = \set {\tuple {B_i, x}: i \in \II, B_i \in \BB, x \in B_i}$

That is, it is the set of ordered pairs of which the first coordinate is a set $B_i \in \BB$ and the second coordinate is an element of $B_i$.

This is, by construction, a subset of the cartesian product:

$\CC \subseteq \ds \BB \times \bigcup_{i \mathop \in \II} B_i$

By hypothesis all sets $B$ are non-empty.

Thus there exists at least one $\tuple {B_i, x}$ element in $\CC$ for each $B_i \in \CC$.

By Equality of Ordered Pairs, if $B_j \ne B_k$ in $\BB$, then $\tuple {B_i, x} \ne \tuple {B_j, x}$ for all such pairs in $\CC$.

So any two sets $\set {\tuple {B_j, x}: x \in B_j}$ and $\set {\tuple {B_k, x} :x \in B_k}$ are disjoint, by the inequality of their first coordinates.

Then $\CC$ is an indexed family of disjoint non-empty sets in $\BB \times \bigcup_{i \mathop \in \II} B_i$.

Hence $\CC$ satisfies the hypotheses of the third formulation of the Axiom of Choice.

Let $c$ be a set satisfying:

$\forall r \in \CC: c \cap r = \set t$

All elements of $c$ are ordered pairs:

the first coordinate of which is a set in $B_i \in \BB$

and:

the second coordinate of which is an element $x \in B_i$.

Viewed as a relation, each set in $\BB$ bears $c$ to an element in that set.

Every set in $\BB$ bears $c$ to exactly one element inside that set.

So $c$ is in fact a mapping and satisfies the criteria of a choice function as expounded in formulation $(1)$.

$\Box$

Formulation 1 iff Formulation 4

We note from Set equals Union of Power Set that:

$x = \ds \map \bigcup {\powerset x}$

Setting $\powerset A =: s$, we see that from Formulation $1$:

$\ds \paren {\O \notin \powerset A \implies \exists \paren {f: \powerset A \to \bigcup \powerset A}: \forall x \in \powerset A: \map f x \in x}$

That is:

for every non-empty proper subset of $A$

there exists a mapping $f: \powerset A \to A$ such that:

for every non-empty proper subset $x$ of $A$: $\map f x \in x$.

That is Formulation $4$ of the Axiom of Choice

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents