# Additional axioms ## Axiom of choice > *Axiom*: let $C$ be a collection of nonempty sets. Then there exists a map > >$$ > f: C \to \bigcap_{A \in C} A >$$ > > with $f(A) \in A$. > > * The image of $f$ is a subset of $\bigcap_{A \in C} A$. > * The function $f$ is called a **choice function**. The following statements are equivalent to the axiom of choice. * For any two sets $A$ and $B$ there does exist a surjective map from $A$ to $B$ or from $B$ to $A$. * The cardinality of an infinite set $A$ is equal to the cardinality of $A \times A$. * Every vector space has a basis. * For every surjective map $f: A \to B$ there is a map $g: B \to A$ with $f(g(b)) = b$ for all $b \in B$. ## Axiom of regularity > *Axiom*: let $X$ be a nonempty set of sets. Then $X$ contains an element $Y$ with $X \cap Y = \varnothing$. As a result of this axiom any set $S$ cannot contain itself.