Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object . Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing. Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least of them), identified with subsets of the set of natural numbers, that were not there in the old universe, and thereby violate the continuum hypothesis. In order to intuitively justify such an expansion, it is best to think of the "old universe" as a model of the set theory, which is itself a set in the "real universe" . By the Löwenheim–Skolem theorem, can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in ) of that are not in . Specifically, there is an ordinal that "plays the role of the cardinal " in , but is actually countable in . Working in , it should be easy to find one distinct subset of per each element of . (For simplicity, this family of subsets can be characterized with a single subset .) However, in some sense, it may be desirable to "construct the expanded model within ". This would help ensure that "resembles" in certain aspects, such as being the same as (more generally, that cardinal collapse does not occur), and allow fine control over the properties of .