<logic> If "S-A" denotes system S minus axiom A, then for many logicians an axiom A is independent of system S iff neither A nor ~A are theorems of S-A. For some, A is independent iff A is not a theorem of S-A, even if ~A is a theorem of S-A. The former concept is equivalent to the undecidability of wff A in S.
[Glossary of First-Order Logic]
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