force, illocutionary See PHILOSOPHY OF LANGUAGE,. SPEECH ACT THEOR. forcing, a method introduced by Paul J. Cohen – see his Set Theory and the Continuum Hypothesis (1966) – to prove independence results in Zermelo-Fraenkel set theory (ZF). Cohen proved the independence of the axiom of choice (AC) from ZF, and of the continuum hypothesis (CH) from ZF ! AC. The consistency of AC with ZF and of CH with ZF ! AC had previously been proved by Gödel by the method of constructible sets. A model of ZF consists of layers, with the elements of a set at one layer always belonging to lower layers. Starting with a model M, Cohen’s method produces an ‘outer model’ N with no more levels but with more sets at each level (whereas Gödel’s method produces an ‘inner model’ L): much of what will become true in N can be ‘forced’ from within M. The method is applicable only to hypotheses in the more ‘abstract’ branches of mathematics (infinitary combinatorics, general topology, measure theory, universal algebra, model theory, etc.); but there it is ubiquitous. Applications include the proof by Robert M. Solovay of the consistency of the measurability of all sets (of all projective sets) with ZF (with ZF ! AC); also the proof by Solovay and Donald A. Martin of the consistency of Martin’s axiom (MA) plus the negation of the continuum hypothesis (-CH) with ZF ! AC. (CH implies MA; and of known consequences of CH about half are implied by MA, about half refutable by MA ! -CH.) Numerous simplifications, extensions, and variants (e.g. Boolean-valued models) of Cohen’s method have been introduced. See also INDEPENDENCE RESULTS, SET THEOR. J.Bur.