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.