Forcing With Copies of Uncountable Ordinals

Abstract

For a relational structure $\mathbb{X}$ we investigate the partial order $⟨\mathbb{P}(\mathbb{X}),\subset ⟩$, where $\mathbb{P}(\mathbb{X}):=\{f[X]:f\in \mathrm{Emb}(\mathbb{X})\}$. Here we consider uncountable ordinals. Since $\mathrm{sq}\mathbb{P}(\alpha )$ is isomorphic to the direct product ${\prod }_{i=1}^n(\mathrm{sq}\mathbb{P}({\omega }^{{\delta }_i}){)}^{s_i}$, where $\alpha ={\omega }^{{\delta }_n}{s}_n+\cdots +{\omega }^{{\delta }_1}{s}_1+m$ 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 $\mathrm{sq}\mathbb{P}(\alpha )$ is $\sigma$-closed and completely embeds $P(\omega )/\mathrm{Fin}$ and, hence, preserves ${\omega }_1$ and forces $|\mathfrak{c}|=|\mathfrak{h}|$, or, otherwise, completely embeds the algebra $P(\lambda )/[\lambda {]}^{<\lambda }$, for some regular $\omega <\lambda \le \mathrm{cf}(\delta )$, and collapses ${\omega }_2$ to $\omega$. Regarding the Cantor normal form, the first case appears iff for each $i\le n$ we have $\mathrm{cf}({\delta }_i)\le \omega$, or ${\delta }_i={\theta }_i+\mathrm{cf}({\delta }_i)$, where $\mathrm{Ord}\ni {\theta }_i\ge \mathrm{cf}({\delta }_i)>\mathrm{cf}({\theta }_i)=\omega$ and ${\theta }_i={\lim }_{n\to \omega }{\delta }_n$, where $\mathrm{cf}({\delta }_n)=\mathrm{cf}({\delta }_i)$, for all $n\in \omega$.

22 pages