ramified forcing
In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by {{harvtxt|Cohen|1963}} to prove the independence of the continuum hypothesis from Zermelo–Fraenkel set theory. Ramified forcing starts with a model {{mvar|M}} of set theory in which the axiom of constructibility, {{math|1=V = L}}, holds, and then builds up a larger model {{math|M[G]}} of Zermelo–Fraenkel set theory by adding a generic subset {{mvar|G}} of a partially ordered set to {{mvar|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 {{math|Vα}} for ordinals {{math|α}}. Their simplification was originally called "unramified forcing" {{harv|Shoenfield|1971}}, but is now usually just called "forcing". As a result, ramified forcing is only rarely used.
References
{{refbegin}}
- {{Citation | last1=Cohen | first1=P. J. | authorlink=Paul Cohen (mathematician)|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 | year=1963 | journal=Proceedings of the National Academy of Sciences of the United States of America | volume=50 | issue=6 | pages=1143–1148 | doi=10.1073/pnas.50.6.1143 | pmid=16578557 | pmc=221287 | issn=0027-8424 | jstor=71858| bibcode=1963PNAS...50.1143C | doi-access=free }}.
- {{citation
|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.
|mr=0280359}}.
{{refend}}
Category:Forcing (mathematics)
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