# iterated forcing

We can define an iterated forcing of length $\alpha$ by induction as follows:

Let $P_{0}=\emptyset$.

Let $\hat{Q}_{0}$ be a forcing notion.

For $\beta\leq\alpha$, $P_{\beta}$ is the set of all functions $f$ such that $\operatorname{dom}(f)\subseteq\beta$ and for any $i\in\operatorname{dom}(f)$, $f(i)$ is a $P_{i}$-name for a member of $\hat{Q}_{i}$. Order $P_{\beta}$ by the rule $f\leq g$ iff $\operatorname{dom}(g)\subseteq\operatorname{dom}(f)$ and for any $i\in\operatorname{dom}(f)$, $g\upharpoonright i\Vdash f(i)\leq_{\hat{Q}_{i}}g(i)$. (Translated, this means that any generic subset including $g$ restricted to $i$ forces that $f(i)$, an element of $\hat{Q}_{i}$, be less than $g(i)$.)

For $\beta<\alpha$, $\hat{Q}_{\beta}$ is a forcing notion in $P_{\beta}$ (so $\Vdash_{P_{\beta}}\hat{Q}_{\beta}$ is a forcing notion).

Then the sequence $\langle\hat{Q}_{\beta}\rangle_{\beta<\alpha}$ is an iterated forcing.

If $P_{\beta}$ is restricted to finite functions that it is called a finite support iterated forcing (FS), if $P_{\beta}$ is restricted to countable functions, it is called a countable support iterated function (CS), and in general if each function in each $P_{\beta}$ has size less than $\kappa$ then it is a $<\kappa$-support iterated forcing.

Typically we construct the sequence of $\hat{Q}_{\beta}$’s by induction, using a function $F$ such that $F(\langle\hat{Q}_{\beta}\rangle_{\beta<\gamma})=\hat{Q}_{\gamma}$.

 Title iterated forcing Canonical name IteratedForcing Date of creation 2013-03-22 12:54:47 Last modified on 2013-03-22 12:54:47 Owner Henry (455) Last modified by Henry (455) Numerical id 5 Author Henry (455) Entry type Definition Classification msc 03E35 Classification msc 03E40 Defines FS Defines CS Defines finite support Defines finite support iterated forcing Defines countable support Defines countable support iterated forcing Defines support iterated forcing