The Feferman–Levy symmetric collapse system

Work over a transitive ground model $V\models \mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{G}\mathrm{C}\mathrm{H}$. The use of Choice is confined to the ground-model aleph sequence and the GCH cardinal calculations below; The Axiom of Choice is not assumed in the eventual symmetric model. Put ${\kappa }_n={\mathrm{\aleph }}_n^V$ for $n<\omega$ and let $P$ be the set of finite partial functions

$p:\omega \times \omega \rightharpoonup {\bigcup }_{n<\omega }{\kappa }_n$

such that $p(n,i)<{\kappa }_n$ whenever $(n,i)\in \mathrm{dom}(p)$, ordered by reverse inclusion. Equivalently, $p$ is a finite set of triples $(n,i,\alpha )$, functional in $(n,i)$, with $\alpha <{\kappa }_n$. Its restriction to the first $m$ layers is

$p\upharpoonright m=\{(n,i,\alpha )\in p:n<m\}\mathrm{.}$

Thus the $n$th layer is the collapse order $\mathrm{Col}(\omega ,{\kappa }_n)$ from Cohen, collapse, and Lévy-collapse forcing orders, and $P$ is their finite-support product.

Let $\mathcal{G}$ consist of the permutations $\pi$ of $\omega \times \omega$ which preserve the first coordinate. Thus $\pi (n,i)=(n,{\pi }_n(i))$ for a sequence of permutations ${\pi }_n\in \mathrm{Sym}(\omega )$. It acts on $P$ by

$\pi p=\{(n,{\pi }_n(i),\alpha ):(n,i,\alpha )\in p\},$

and on names by Automorphisms acting on forcing names. For $m<\omega$ let

$H_m=\{\pi \in \mathcal{G}:{\pi }_n={\mathrm{id}}_{\omega }\text{ for every }n<m\}\mathrm{.}$

The subgroups $H_m$ are normal, $H_{m+1}\subseteq H_m$, and their upward closure is a normal filter $\mathcal{F}$ of subgroups. Hence $(P,\mathcal{G},\mathcal{F})$ is a symmetric system in the sense of Symmetric forcing systems, supports, and hereditarily symmetric names. If $G$ is $V$-generic, its hereditarily symmetric interpretation

$N={\mathrm{H}\mathrm{S}}_{\mathcal{F}}^G$

is called the Feferman–Levy model. A name is said to have $m$-bounded layer support when $H_m$ fixes it.