 The notion of “independence” at issue here is that of non-provability: to say that a given statement is independent of a collection of statements is to say that it is not provable from them, or equivalently that the collection does not logically entail that statement. Consistency, too, is understood in terms of provability: to say that a collection of statements is consistent is to say that no contradiction is provable from it. Hence the two notions, consistency and independence, are inter-definable: a set of statements is consistent if an arbitrarily-chosen contradiction is independent of it, and a statement S is independent of a set C if the set \(C \cup {\sim}S\) is consistent.