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

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.