In mathematical logic, independence is the unprovability of some specific sentence from some specific set of other sentences. The sentences in this set are referred to as "axioms".
A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T. (This concept is unrelated to the idea of "decidability" as in a decision problem.)
A theory T is independent if no axiom in T is provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable.