independence
A well-built axiom system should waste nothing: every axiom must earn its place by stating something the others cannot already prove. An axiom is independent of the rest if it is not a consequence of them — you genuinely need it, because dropping it would leave some of your theorems unprovable. Checking independence is how you know your list of assumptions is lean rather than padded with hidden repetitions. The most celebrated example in all of mathematics is the parallel postulate, which the ancients suspected of being redundant and which turned out, after two millennia, to be perfectly independent.
How do you prove an axiom cannot be derived from the others? You cannot just fail to find a proof — maybe you were not clever enough. Instead you build a model in which all the other axioms are true but the axiom in question is false. Since true axioms can only ever prove true conclusions, the existence of such a model shows the lone axiom cannot follow from the others; if it did, it would be forced true wherever they hold, contradicting your model. To prove the parallel postulate independent, you exhibit a structure — a non-Euclidean model such as the Poincare disk — where incidence, order, congruence, and continuity all hold but a point has more than one parallel through it. That single model settles a 2000-year-old question.
Independence is a quality of elegance and honesty, not of truth. A dependent axiom is not wrong, merely superfluous: you could delete it and prove it back as a theorem. Hilbert devoted careful attention to showing each of his axiom groups is independent of the others, constructing strange custom models for each — geometries violating just one axiom while obeying the rest. The technique is always the same: independence is proved by a model, exactly as consistency is, which is why models are the universal instrument of the axiomatic method.
Is the parallel postulate independent of Euclid's other axioms? Build the Poincare disk: 'points' are inside a circle, 'lines' are arcs perpendicular to its boundary. There all the neutral axioms hold, yet through a point off a line run infinitely many non-meeting lines — the parallel postulate fails. The model exists, so the postulate cannot be a theorem of the others.
One model where every other axiom holds but this one fails — that proves the axiom is independent.
Independence does not mean an axiom is true or false in the real world — only that the others do not force its truth value. A dependent axiom is harmless but redundant; you could prove it as a theorem instead.