Category:Equivalence of Formulations of Axiom of Choice

This category contains pages concerning Equivalence of Formulations of Axiom of Choice:

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 $A$.