Forcing With Copies of Uncountable Ordinals

Abstract

For a relational structure ${\mathbb X}$ we investigate the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X}):=\{ f[X]: f\in \mathop{\rm Emb}\nolimits ({\mathbb X})\}$. Here we consider uncountable ordinals. Since $\mathop{\rm sq}\nolimits {\mathbb P} (\alpha )$ is isomorphic to the direct product $\prod _{i=1}^n (\mathop{\rm sq}\nolimits {\mathbb P} (\omega ^{\delta _i}))^{s_i}$, where $\alpha = \omega ^{\delta _n}s_n+\dots +\omega ^{\delta _1}s_1+ m$ is the Cantor normal form for $\alpha $, the analysis is reduced to the investigation of the posets of the form ${\mathbb P} (\omega ^{\delta })$. It turns out that, in ZFC, either the poset $\mathop{\rm sq}\nolimits {\mathbb P} (\alpha )$ is $\sigma$-closed and completely embeds $P(\omega )/\mathop{\rm Fin}$ and, hence, preserves $\omega _1$ and forces $|{\mathfrak c}|=|{\mathfrak h}|$, or, otherwise, completely embeds the algebra $P(\lambda )/[\lambda ]^{<\lambda }$, for some regular $\omega <\lambda \leq \mathop{\rm cf}\nolimits (\delta )$, and collapses $\omega _2$ to $\omega $. Regarding the Cantor normal form, the first case appears iff for each $i\leq n$ we have $\mathop{\rm cf}\nolimits (\delta _i)\leq \omega $, or $\delta _i = \theta _i + \mathop{\rm cf}\nolimits (\delta _i )$, where $\mathop{\mathrm{Ord}}\nolimits \ni\theta _i \geq \mathop{\rm cf}\nolimits (\delta _i ) >\mathop{\rm cf}\nolimits (\theta _i )=\omega $ and $\theta _i =\lim _{n\rightarrow \omega }\delta _n$, where $\mathop{\rm cf}\nolimits (\delta _n)=\mathop{\rm cf}\nolimits (\delta _i)$, for all $n\in \omega $.