A sentence ? is independent of a given first-order theory T if T neither proves nor refutes ?; that is, it is impossible to prove ? from T, and it is also impossible to prove from T that ? is false. Sometimes, ? is said (synonymously) to be undecidable from T; this is not the same meaning of "decidability" as in a decision problem.
A theory T is independent if each axiom in T is not provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable.
Some authors say that ? is independent of T when T simply cannot prove ?, and do not necessarily assert by this that T cannot refute ?. These authors will sometimes say "? is independent of and consistent with T" to indicate that T can neither prove nor refute ?.
Many interesting statements in set theory are independent of Zermelo-Fraenkel set theory (ZF). The following statements in set theory are known to be independent of ZF, under the assumption that ZF is consistent:
The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo-Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.
The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent.