Independence (mathematical logic)

In mathematical logic, independence is the unprovability of a sentence from other sentences.

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 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.