Independence (mathematical logic)

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.