PlanetMath: iterated forcing

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iterated forcing

(Definition)

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 such that $\operatorname{dom}(f)\subseteq\beta$ and for any $i\in\operatorname{dom}(f)$ , is a -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 restricted to forces that , an element of $\hat{Q}_i$ , be less than .)

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

"iterated forcing" is owned by Henry.

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Also defines:

FS, CS, finite support, finite support iterated forcing, countable support, countable support iterated forcing, support iterated forcing

Attachments:: iterated forcing and composition (Result) by Henry
FS iterated forcing preserves chain condition (Result) by Henry

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Cross-references: size, countable, finite, sequence, forces, restricted, subset, generic, iff, order, functions, forcing, induction, length
There are 10 references to this entry.

This is version 2 of iterated forcing, born on 2002-08-04, modified 2003-01-11.
Object id is 3264, canonical name is IteratedForcing.
Accessed 8670 times total.

Classification:

AMS MSC:	03E35 (Mathematical logic and foundations :: Set theory :: Consistency and independence results)
	03E40 (Mathematical logic and foundations :: Set theory :: Other aspects of forcing and Boolean-valued models)

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