The real layers of the Feferman–Levy model

Let $B=\mathrm{RO}(P)$ and let $B_m$ be its complete subalgebra of $H_m$-fixed values. For $m<\omega$, let $S_m$ be the ground-model set of Boolean names for subsets of $\omega$ of the form

$\dot{x}=\{⟨\check{k},b_k⟩:k<\omega \},b_k\in B_m\mathrm{.}$

In the Feferman–Levy symmetric model $N$, define

$R_m=\{{\dot{x}}_G:\dot{x}\in S_m\}\mathrm{.}$

Equivalently, $R_m$ is the set of reals admitting a Boolean name with $m$-bounded layer support. By Fixed Boolean values come from initial collapse layers, every member of $S_m$ is hereditarily symmetric. Since $H_m$ is normal in the layer-preserving group, every automorphism maps $B_m$ and $S_m$ onto themselves. Therefore the canonical names collecting each $S_m$, and the canonical name for the sequence $⟨R_m:m<\omega ⟩$, are fixed by the whole group and are hereditarily symmetric. In particular every $R_m$ and the displayed sequence are sets of $N$. No choice principle is used in this definition.