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.