Skip to content

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.