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is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by
Solovay and Tennenbaum (
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Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for
Mathematical Logic (47) 2008 pp. 579–606
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Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of
Symbolic Logic (66) 2001, pp. 1865–1883
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Revised countable support iterations of semi-proper forcings are semi-proper and thus preserve
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of forcing notions indexed by some ordinals α, which give a family of
Boolean-valued models
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is not collapsed. This is often accomplished by the use of a preservation theorem such as:
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Solovay, R. M.; Tennenbaum, S. (1971). "Iterated Cohen extensions and
Souslin's problem".
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is often constructed as some sort of limit (such as the direct limit) of the
349:, Perspectives in Mathematical Logic (2 ed.), Berlin: Springer-Verlag,
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218:, can be iterated with appropriate cardinal collapses while preserving
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31:. They also showed that iterated forcing can construct models where
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Eisworth, Todd; Moore, Justin Tatch (2009), Milovich, David (ed.),
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Countable support iterations of proper forcings are proper (see
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A key consideration is that, typically, it is necessary that
27:) in their construction of a model of set theory with no
304:, Springer Monographs in Mathematics, Berlin, New York:
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holds and the continuum is any given regular cardinal.
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Shelah, S., Proper and
Improper Forcing, Springer 1992
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In iterated forcing, one has a transfinite sequence
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151:Fundamental Theorem of Proper Forcing
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119:) are c.c.c. and thus preserve
302:Set Theory: Millennium Edition
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347:Proper and improper forcing
245:using methods developed by
238:{\displaystyle \omega _{1}}
203:{\displaystyle \omega _{1}}
173:{\displaystyle \omega _{1}}
139:{\displaystyle \omega _{1}}
104:{\displaystyle \omega _{1}}
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63:using a forcing notion in
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29:Suslin tree
253:References
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375:. 2.
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40:P
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