Independent axiom set
= Independent axiom set
For an <independent axiom set>, <semantic entailment> of any one axiom from the remaining <axioms> fails. By the completeness theorem, this is equivalent to no member being formally derivable from the others in a complete calculus. To establish independence it suffices to give, for each axiom, a <first-order structure> satisfying all the other <axioms> and failing that one.