Ramified forcing

In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by Cohen (1963) to prove the independence of the continuum hypothesis from Zermelo–Fraenkel set theory. Ramified forcing starts with a model M of set theory in which the axiom of constructibility, V = L, holds, and then builds up a larger model M[G] of Zermelo–Fraenkel set theory by adding a generic subset G of a partially ordered set to M, imitating Kurt Gödel's constructible hierarchy.

Dana Scott and Robert Solovay realized that the use of constructible sets was an unnecessary complication, and could be replaced by a simpler construction similar to John von Neumann's construction of the universe as a union of sets R(α) for ordinals α. Their simplification was originally called "unramified forcing" (Shoenfield 1971), but is now usually just called "forcing". As a result, ramified forcing is only rarely used.

References


This article is issued from Wikipedia - version of the Wednesday, February 15, 2012. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.