Ramified forcing

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In mathematics, ramified forcing is the original form of forcing introduced by Cohen (1963). Ramified forcing starts with a model M of V = L, and builds up larger model M[G] of ZF by adding a generic subset G of a poset to M, by imitating Godel's constructible hierarchy. Scott and Solovay realized that the use of constructible sets was an unnecessary complication, and could be replaced by a simpler construction similar to von Neumann's construction of the universe as a union of sets R(α) for ordinals α. (This simplification was originally called "unramified forcing" (Schoenfield 1971), but is now usually just called "forcing".) As a result, ramified forcing is only rarely used.

[edit] References

• Cohen, P. J. (1966), Set Theory and the Continuum Hypothesis, Menlo Park, CA: W. A. Benjamin 
• Cohen, Paul J. (1963), “The Independence of the Continuum Hypothesis”, Proceedings of the National Academy of Sciences of the United States of America 50 (6): 1143–1148, ISSN 0027-8424, <http://links.jstor.org/sici?sici=0027-8424%2819631215%2950%3A6%3C1143%3ATIOTCH%3E2.0.CO%3B2-5> 
• Shoenfield, J. R. (1971), “Unramified forcing”, Axiomatic Set Theory, vol. XIII, Part I, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 357--381, MR0280359