port from mathematics-physics notes

docs/mathematics/set-theory/additional-axioms.md

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+# 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.