Suppose that $(\mathbb{P}_{\alpha}, \underset{\sim}{\mathbb{Q_{\alpha}}} : \alpha<\beta)$ is a forcing iteration, and that for each $\alpha$, there is a name $\underset{\sim}{\eta_{\alpha}}$ for the generic real added at the $\alpha$th stage. Denote the limit of the iteration by $\mathbb{P}_{\beta}$. Does it follow that $\{\eta_{\alpha} : \alpha<\beta\}$ is generic for $\mathbb{P}_{\beta}$? Intuitively it seems true, but I don't know how to prove it.
Generic reals in forcing iterations
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logic
set-theory
forcing
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0In this generality the result is false $\{\eta_\alpha: \alpha<\beta\}$ is not even a filter (in general). I suppose you have in mind some forcing notions where the generic real code the generic filter like in Cohen, Sacks, Laver forcings... – 2012-03-28
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0Yes, of course. You may assume that we're dealing with a nice situation where the generic filter is coded by the generic real. – 2012-03-28