Iterated Forcing

In mathematics, iterated forcing 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 in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcing can construct models where Martin's axiom holds and the continuum is any given regular cardinal.

In iterated forcing, one has a transfinite sequence ''P''_α of forcing notions indexed by some ordinals α, which give a family of Boolean-valued models ''V''^''P''_α. If α+1 is a successor ordinal then ''P''_α+1 is often constructed from ''P''_α using a forcing notion in ''V''^''P''_α, while if α is a limit ordinal then ''P''_α is often constructed as some sort of limit (such as the direct limit) of the ''P''_β for β<α.

A key consideration is that, typically, it is necessary that $\omega_1$ is not collapsed. This is often accomplished by the use of a preservation theorem such as:
* Finite support iteration of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve $\omega_1$.
* Countable support iterations of proper forcings are proper (see Fundamental Theorem of Proper Forcing) and thus preserve $\omega_1$.
* Revised countable support iterations of semi-proper forcings are semi-proper and thus preserve $\omega_1$.

Some non-semi-proper forcings, such as Namba forcing, can be iterated with appropriate cardinal collapses while preserving $\omega_1$ using methods developed by Saharon Shelah.Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865–1883

References

Sources

*
*
*
*

External links

*{{citation, url=http://www.math.cmu.edu/~eschimme/Appalachian/EisworthMooreNotes.pdf, title=ITERATED FORCING AND THE CONTINUUM HYPOTHESIS, series=Appalachian Set Theory Workshop lecture notes , year=2009, first= Todd , last=Eisworth , first2= Justin Tatch, last2= Moore, editor-first= David , editor-last=Milovich

Forcing (mathematics)