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Forcing by non-scattered sets. (English) Zbl 1250.03102
Summary: We show that for each non-scattered linear order $$\langle L,<\rangle$$ the set of non-scattered subsets of $$L$$ ordered by the inclusion is forcing-equivalent to the two-step iteration of the Sacks forcing and a $$\sigma$$-closed forcing. If the equality $$\mathrm{sh} (\mathbb S) =\aleph_1$$ or PFA holds in the ground model, then the second iterand is forcing-equivalent to the algebra $$P(w)/\mathrm{Fin}$$ of the Sacks extension.

##### MSC:
 03E40 Other aspects of forcing and Boolean-valued models 03E35 Consistency and independence results 06A05 Total orders
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##### References:
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