Forcing (mathematics)

In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. It was first used, in 1962, to prove the independence of the continuum hypothesis and the axiom of choice. Forcing was considerably reworked and simplified in the sixties, and is nowadays a basic technique.

Related Topics:
Axiomatic set theory - Paul Cohen - Zermelo-Fraenkel axioms - 1962 - Continuum hypothesis - Axiom of choice - Sixties

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Forcing is equivalent to the method of Boolean-valued models, which is conceptually more natural and intuitive, but usually much more difficult to apply.

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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 weren't 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×{0,1}, identify x∈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.

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