The Good-Tree-Watson symmetric Stone model

Work in a transitive ZFC ground model $M$ with GCH (The Axiom of Choice), and fix a regular uncountable cardinal $\lambda$. Put $P:=\mathrm{Fn}(\lambda \times \mathbb{R}\times \lambda \times \lambda ,2,\lambda ),$ the set of partial functions with domain a subset of $\lambda \times \mathbb{R}\times \lambda \times \lambda$ of size below $\lambda$ and values in $2$, ordered by reverse inclusion: $q\le p$ means $p\subseteq q$ (Forcing preorders, compatibility and filters).

Fix an $M$-generic filter $H\subseteq P$ in an ambient universe. Let $B:=\mathrm{RO}(P)$ be the regular-open completion (Completeness, regular opens, and order continuity, Choice-free regular open completion of forcing preorders).

The group. Put $J=\lambda \times \mathbb{R}\times \lambda$. Use all permutations $\pi$ of $J$ of the form $\pi (\xi ,r,\alpha )=(\xi ,{\rho }_{\xi }(r),{\sigma }_{\xi ,r}(\alpha )),{\rho }_{\xi }(r)={\epsilon }_{\xi }r+t_{\xi },$ where ${\epsilon }_{\xi }\in \{-1,1\}$, $t_{\xi }\in \mathbb{R}$, and each ${\sigma }_{\xi ,r}$ is a permutation of $\lambda$. All these data belong to $M$. The real isometries are allowed independently for different $\xi$. Composition replaces two real maps by $r\mapsto {\epsilon }^{\mathrm{\prime }}\epsilon r+{\epsilon }^{\mathrm{\prime }}t+t^{\mathrm{\prime }}$ in each component; inverses have the same form. The fibre permutations compose with the corresponding reindexing of $r$, and their inverses are permutations too. Thus these maps form a group, containing translations as well as reflections.

The induced action on conditions fixes the last coordinate: $(\pi p)(\pi (\xi ,r,\alpha ),\delta )=p(\xi ,r,\alpha ,\delta )\mathrm{.}$ It preserves domain cardinalities and reverse inclusion and has the stated inverse, hence acts by forcing automorphisms. It also acts on $B$ by taking images of regular open sets: an order automorphism is a homeomorphism for the downward-open topology and commutes with interior and closure. Write $G$ for this group of induced automorphisms. Its action on $P$-names is the recursion of Automorphisms acting on forcing names.

The filter and interpretation. For $e\subseteq J$ of ground-model size less than $\lambda$, put $\mathrm{fix}(e)=\{\pi \in G:\pi {\mathrm{\mid }}_e=\mathrm{i}\mathrm{d}\}$. Let $\mathcal{F}$ consist of the subgroups containing some such stabilizer. It is upward closed, contains $G=\mathrm{fix}(\mathrm{\varnothing })$, and $\mathrm{fix}(e\cup f)=\mathrm{fix}(e)\cap \mathrm{fix}(f)$. Also $\pi \mathrm{fix}(e){\pi }^{-1}=\mathrm{fix}(\pi [e])$ and $\mathrm{\mid }\pi [e]\mathrm{\mid }=\mathrm{\mid }e\mathrm{\mid }$, proving normality. For fewer than $\lambda$ subgroups in $\mathcal{F}$, ground-model AC chooses their support witnesses; regularity of $\lambda$ makes their union have size less than $\lambda$, and its stabilizer is contained in their intersection. Thus $\mathcal{F}$ is $<\lambda$-complete in $M$.

Define $N={\mathrm{H}\mathrm{S}}_{\mathcal{F}}^H$ using the symmetric system $(P,G,\mathcal{F})$ (Symmetric forcing systems, supports, and hereditarily symmetric names). It is a transitive ZF model with $M\subseteq N\subseteq M[H]$ (Hereditarily symmetric interpretations form a transitive ZF model).

The canonical families. For $\xi <\lambda$, $r\in \mathbb{R}$ and $\alpha <\lambda$ let $x_{\xi r\alpha }:=\{\delta <\lambda :(\mathrm{\exists }p\in H)p(\xi ,r,\alpha ,\delta )=1\}\mathrm{.}$ Thus $x_{\xi r\alpha }$ is a generic subset of $\lambda$, not in general a real. Let $X_{\xi r}:=\{x_{\xi r\alpha }:\alpha <\lambda \}$, let $R_{\xi }:=\{X_{\xi r}:r\in \mathbb{R}\}$, and let $\mathcal{R}:=\{R_{\xi }:\xi <\lambda \}$, keeping $M$ for the ground model. For two distinct triples and any condition, choose a last coordinate $\delta <\lambda$ unused at both triples and extend the condition by opposite values there. This is possible because fewer than $\lambda$ coordinates have been used. These extensions are dense, so genericity makes all the canonical subsets for distinct triples distinct. In particular the nonempty families $X_{\xi r}$ for distinct $(\xi ,r)$ are disjoint. Consequently the rule $d_{\xi }(X_{\xi r},X_{\xi s}):=\frac{\mathrm{\mid }r-s\mathrm{\mid }}{1+\mathrm{\mid }r-s\mathrm{\mid }}$ is a well-defined metric on $R_{\xi }$, transported from the displayed bounded metric on $\mathbb{R}$; no metric is induced from the sets $x_{\xi r\alpha }\subseteq \lambda$ themselves.

The name for $x_{\xi r\alpha }$ is fixed by $\mathrm{fix}(\{(\xi ,r,\alpha )\})$, the name for $X_{\xi r}$ is fixed by $\mathrm{fix}(\{(\xi ,r,{\alpha }_0)\})$ for any fixed ${\alpha }_0<\lambda$, and the names for $R_{\xi }$ and $\mathcal{R}$ are fixed by the whole group. Their members are hereditarily symmetric by induction, so all four kinds of canonical object lie in $N$. The component family has the full index set $\lambda$; this does not assert that the real-label enumeration of each component is in $N$. The canonical name for $d_{\xi }$ is fixed by the whole group since $\mathrm{\mid }{\rho }_{\xi }(r)-{\rho }_{\xi }(s)\mathrm{\mid }=\mathrm{\mid }r-s\mathrm{\mid }$; its subnames are hereditarily symmetric by the same calculation, so $d_{\xi }\in N$. The metric triangle inequality follows from the real triangle inequality and, for $a,b\ge 0$, $\frac{a+b}{1+a+b}\le \frac{a}{1+a}+\frac{b}{1+b}$; the function $t\mapsto t\mathrm{/}(1+t)$ is increasing for $t\ge 0$. Separation and symmetry follow directly from the distinct real labels. Thus the metric assertion is justified internally as well as in the full extension.

The case $\lambda ={\omega }_1$ is the case used for the dependent-choice model below; the same definition with larger $\lambda$ is the one whose $<\lambda$-sequence closure is proved in the next item.

• Why the presentation is indexed by $\lambda$ and not by $\omega$. The paper's Theorems 1–3 use the countable index set and finite supports, and the paragraph after Theorem 3 states the regular-$\lambda$ replacement with supports of size below $\lambda$. The dependent-choice model needs the replacement with supports of size $<\lambda$. Closure under sequences is not inferred here from the finite-support presentation; it is a separate proof obligation for the regular-$\lambda$ construction.

• What is not claimed here. The definition does not assert dependent choice, the failure of Stone's theorem, or the existence of the metric sum of the components; those are separate items of this page.

• Source convention repaired. The source's Theorem 1 proof describes the real actions as identity or reflection, a class not closed under composition: two distinct reflections compose to a nonzero translation. Its Claim 1.3 also uses a reflection on just one component. The componentwise affine isometry group above makes closure and this independence explicit, preserves the displayed metric, and retains all the canonical support calculations.