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Forcing with copies of countable ordinals. (English) Zbl 1386.03065
Summary: Let \( \alpha \) be a countable ordinal and \( \mathbb{P}(\alpha )\) the collection of its subsets isomorphic to \( \alpha \). We show that the separative quotient of the poset \( \langle \mathbb{P}(\alpha ), \subset \rangle \) is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form \( P(\omega ^\gamma )/\mathcal {I}_{\omega ^\gamma }\), where \( \gamma \in \mathrm {Lim}\cup \{ 1 \}\) and \( \mathcal {I}_{\omega ^\gamma }\) is the corresponding ordinal ideal. Moreover, the poset \( \langle \mathbb{P} (\alpha ), \subset \rangle \) is forcing equivalent to a two-step iteration of the form \( (P(\omega )/\mathrm {Fin})^+ \ast \pi \), where \( [\omega ] \Vdash \) “\( \pi \) is an \( \omega _1\)-closed separative pre-order” and, if \( \mathfrak{h}=\omega _1\), to \( (P(\omega )/\mathrm {Fin})^+\). Also we analyze the quotients over ordinal ideals \( P(\omega ^\delta )/\mathcal {I}_{\omega ^\delta }\) and the corresponding cardinal invariants \( \mathfrak{h}_{\omega ^\delta }\) and \( \mathfrak{t}_{\omega ^\delta }\).

MSC:
03E40 Other aspects of forcing and Boolean-valued models
03E10 Ordinal and cardinal numbers
03C15 Model theory of denumerable and separable structures
03E35 Consistency and independence results
06A06 Partial orders, general
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