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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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