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 Pα of forcing notions indexed by some ordinals α, which give a family of Boolean-valued models VPα. If α+1 is a successor ordinal then Pα+1 is often constructed from Pα using a forcing notion in VPα, while if α is a limit ordinal then Pα is often constructed as some sort of limit (such as the direct limit) of the Pβ 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.