Consider $D$ be a filter over $\omega$ and $\mathbb{L}(D)$ be the Laver forcing (i,e, the elements of $\mathbb{L}(D)$ are trees $T\subseteq\omega^{<\omega}$ with stem $s_T$ and $\forall s\in LV_n(T)(\{k\in\omega:s\cup\{(n,k)\}\in T\}\in D)$)
I'm reading the paper "on the cofinality of the splitting number" by Dow and Shelah and in such paper appears the following definition: Take $E\subset\mathbb{L}(D)$ be a dense set and define $\rho_E:\omega^{<\omega}\rightarrow\omega_1$ as $\rho_E(s)=0$ iff there exists $T\in E$ such that $s=s_T$ and $\rho_E(s)\leq\alpha$ iff $\{k\in\omega:\rho_E(s\hat \ k)<\alpha \ \}\in D^+$ (Here $D^+$ denotes the positive elements of $D$, i,e, the sets $F\in[\omega]^\omega$ which meets every element of $D$).
I have many doubts about such construction. First, why $\rho_E[\omega^{<\omega}]\subset\omega_1$? which is the intuition about this definition? How can be building this function?
I apreciate any comment which help me to clarify my mind. Thanks for advance.
(For simplicity, I'll write "$\sigma k$" for "$\sigma\cup(\vert\sigma\vert, k)$.")
Let $X=\rho_E[\omega^{<\omega}]$. I claim that $X$ is closed downwards; from this the desired result will of course follow.
Suppose otherwise. Fix $\theta_0<\theta_1$ with $\theta_0\not\in X$ and $\theta_1=\min(X\setminus\theta_0)$, and choose a $\sigma$ such that $\rho_E(\sigma)=\theta_1$. Then a fortiori $\rho_E(\sigma)\le\theta_1$, and so we must have $$\{k: \rho(\sigma k)<\theta_1\}\in D^+.$$ But by definition of $\theta_1$, we have that $\rho_E(\tau)<\theta_1\implies\rho_E(\tau)<\theta_0$ for every $\tau$, and so $$\{k: \rho(\sigma k)<\theta_0\}\in D^+.$$ But this means $\rho_E(\sigma)\le\theta_0<\theta_1=\rho_E(\sigma)$, a contradiction. Note that no properties of $E,D$, or $D^+$ are used here - this is a general argument that any reasonable way of "ranking" strings must have downwards-closed range, and consequently range properly contained in $\omega_1$.