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Independence (mathematical logic)
In mathematical logic, independence refers to 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.
==Usage note==
Some authors say that σ is independent of ''T'' if ''T'' simply cannot prove σ, and do not necessarily assert by this that ''T'' cannot refute σ. These authors will sometimes say "σ is independent of and consistent with ''T''" to indicate that ''T'' can neither prove nor refute σ.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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