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# 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 $F(\langle\hat{Q}_{\beta}\rangle_{{\beta<\gamma}})=\hat{Q}_{\gamma}$.

## Mathematics Subject Classification

03E35*no label found*03E40

*no label found*

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