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

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, doi:10.1073/pnas.50.6.1143, ISSN 0027-8424, JSTOR 71858, PMC 221287, PMID 16578557 .
Shoenfield, J. R. (1971), "Unramified forcing", Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 357–381, MR 0280359 .

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