Forcing (mathematics)

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In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1962, to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory. Forcing was considerably reworked and simplified in the 1960s, and has proven to be an extremely powerful technique both within set theory and in other areas of mathematical logic such as recursion theory.

Descriptive set theory uses both the notion of forcing from recursion theory as well as set theoretic forcing. Forcing has also been used in model theory but it is common in model theory to define genericity directly without mention of forcing.

Contents

Intuitions

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 V to a larger universe V*. In this bigger universe, for example, one might have lots of new subsets of ω = {0,1,2,…} 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 "paradoxes" about infinity. In principle, one could consider $V^* = V \times \{0,1\}$, identify $x \in V$with (x,0), and then introduce an expanded membership relation involving the "new" sets of the form (x,1). Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe.