Ciro Santilli A set of axioms is consistent if they don't lead to any contradictions.
When a set of axioms is not consistent, false can be proven, and then everything is true, making the set of axioms useless.
A theorem is said to be independent from a set of axioms if it cannot be proven neither true nor false from those axioms.
It or its negation could therefore be arbitrarily added to the set of axioms.