Common axioms defined for binary relations.

• completeness
• weakly connected
• transitivity
• negative transitive
• reflexivity
• irreflixivity
• symmetric
• asymmetric
• anitsymmetric
• acyclic

axioms inconsistent => There exists x,y in X s.t. axioms do not hold

axioms not independent => a (strict) subset of the axioms => the axioms

The axiom of choice:

• given any set M there exists a "choice function" f s.t. f(A) in A subset of M forall nonempty A subset of M.

The axiom of choice is subtle and powerful.