Ramified forcing

Ramified forcing

In mathematics, ramified forcing is the original form of forcing introduced by harvtxt|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" harv|Schoenfield|1971, but is now usually just called "forcing".) As a result, ramified forcing is only rarely used.

References

*Citation | last1=Cohen | first1=P. J. | title=Set Theory and the Continuum Hypothesis | publisher=W. A. Benjamin | location=Menlo Park, CA | year=1966
*Citation | last1=Cohen | first1=Paul J. | title=The Independence of the Continuum Hypothesis | url=http://links.jstor.org/sici?sici=0027-8424%2819631215%2950%3A6%3C1143%3ATIOTCH%3E2.0.CO%3B2-5 | year=1963 | month=15 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=50 | issue=6 | pages=1143–1148
*citation|id=MR|0280359
last=Shoenfield|first= J. R.
chapter=Unramified forcing|year= 1971 |title=Axiomatic Set Theory |series=Proc. Sympos. Pure Math.|volume= XIII, Part I|pages= 357--381 |publisher=Amer. Math. Soc.|publication-place= Providence, R.I.


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