independence results proofs of non-deducibility. Any of the following equivalent conditions may be called independence: (1) A is not deducible from B; (2) its negation – A is consistent with B; (3) there is a model of B that is not a model of A; e.g., the question of the non-deducibility of the parallel axiom from the other Euclidean axioms is equivalent to that of the consistency of its negation with them, i.e. of non-Euclidean geometry. Independence results may be not absolute but relative, of the form: if B is consistent (or has a model), then B together with – A is (or does); e.g. models of non-Euclidean geometry are built within Euclidean geometry. In another sense, a set B is said to be independent if it is irredundant, i.e., each hypothesis in B is independent of the others; in yet another sense, A is said to be independent of B if it is undecidable by B, i.e., both independent of and consistent with B. The incompleteness theorems of Gödel are independence results, prototypes for many further proofs of undecidability by subsystems of classical mathematics, or by classical mathematics as a whole, as formalized in Zermelo- Fraenkel set theory with the axiom of choice (ZF ! AC or ZFC). Most famous is the undecidability of the continuum hypothesis, proved consistent relative to ZFC by Gödel, using his method of constructible sets, and independent relative to ZFC by Paul J. Cohen, using his method of forcing. Rather than build models from scratch by such methods, independence (consistency) for A can also be established by showing A implies (is implied by ) some A* already known independent (consistent). Many suitable A* (Jensen’s Diamond, Martin’s Axiom, etc.) are now available. Philosophically, formalism takes A’s undecidability by ZFC to show the question of A’s truth meaningless; Platonism takes it to establish the need for new axioms, such as those of large cardinals. (Considerations related to the incompleteness theorems show that there is no hope even of a relative consistency proof for these axioms, yet they imply, by way of determinacy axioms, many important consequences about real numbers that are independent of ZFC.) With non-classical logics, e.g. second-order logic, (1)–(3) above may not be equivalent, so several senses of independence become distinguishable. The question of independence of one axiom from others may be raised also for formalizations of logic itself, where many-valued logics provide models. See also FORCING, GÖDEL’s INCOMPLETE- NESS THEOREMS , SET THEOR. J.Bur.