Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique discovered by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. Forcing was 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 notion 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 equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.

Intuitively, forcing consists of expanding the set theoretical universe ${\displaystyle V}$ to a larger universe ${\displaystyle V^{*}}$. In this bigger universe, for example, one might have lots of new subsets of ${\displaystyle \omega }$ ${\displaystyle =\{0,1,2,\ldots \}}$ that were not there in the old universe, and thereby violate the continuum hypothesis. While impossible on the face of it, this is just another version of Cantor's paradox about infinity. In principle, one could consider

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