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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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