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 ÃÂ is independent of a given first-order theory T if T neither proves nor refutes ÃÂ; that is, it is impossible to prove ÃÂ from T, and it is also impossible to prove from T that ÃÂ is false. Sometimes, ÃÂ 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.
Some authors say that ÃÂ is independent of T when T simply cannot prove ÃÂ, and do not necessarily assert by this that T cannot refute ÃÂ. These authors will sometimes say "ÃÂ is independent of and consistent with T" to indicate that T can neither prove nor refute ÃÂ.
Many interesting statements in set theory are independent of ZermeloâÂÂFraenkel set theory (ZF). The following statements in set theory are known to be independent of ZF, under the assumption that ZF is consistent:
The following statements (none of which have been proved false) cannot be proved in ZFC (the ZermeloâÂÂFraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.
The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent.
A set of sentences is independent, or simply independent, if no sentence in the set is provable from the others. This is equivalent to saying that for each sentence in the set there exists an interpretation under which that sentence is false but all the others are true.
There is another sense of independence, called complete independence, or strong independence. A set of sentences is completely independent if for every subset, there exists an interpretation under which all the members of that subset are true and all the others are false. This is equivalent to saying that all combinations of the sentences being true or false are consistent.
Since 2000, logical independence has become understood as having crucial significance in the foundations of physics.