Forcing with copies of countable ordinals

Abstract:

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 $\mathfrak {t}_{\omega ^\delta }$.