Cofinality and size of the scale subspace

Fix $b\in Q$. Recursively for $\xi <{\omega }_1$ construct indices ${\alpha }_{\xi }<\lambda$ and functions $g_{\xi }\in Q$. Given the earlier choices, put $t_{\xi }(n)=\sup (\{b(n)+1\}\cup \{g_{\eta }(n)+1:\eta <\xi \})$. All entries in this supremum are below ${\mathrm{\aleph }}_n$, since that cardinal is a limit and the earlier functions are in $Q$. The set is countable, so F3–F4 give $t_{\xi }(n)<{\mathrm{\aleph }}_n$ and $t_{\xi }\in Q$. The countably many earlier indices are bounded in regular $\lambda$ by F3–F4. By F1 there is an index above them all with $t_{\xi }{<}^{\ast }{f}_{\alpha }$: first obtain eventual domination by cofinality, and if necessary pass to a larger scale index, whose strict eventual increase preserves domination. Choose the least such index ${\alpha }_{\xi }$ and set $g_{\xi }(n)=\max (t_{\xi }(n),f_{{\alpha }_{\xi }}(n))$. It belongs to $Q$, is eventually equal to $f_{{\alpha }_{\xi }}$, and is strictly above $b$ and every earlier $g_{\eta }$ at every coordinate. The exceptional modification set is contained in the finite set where $t_{\xi }(n)\ge f_{{\alpha }_{\xi }}(n)$. These are specified rules with proved witnesses, so F7 constructs them throughout ${\omega }_1$. The initial stage uses only $b(n)+1$ and has no earlier-index obligation.

Fix $\alpha <\lambda$. Every point eventually equal to $f_{\alpha }$ is determined by a finite exceptional set $S\subseteq B$ and its values in ${\prod }_{n\in S}({\mathrm{\aleph }}_n+1)$. There are countably many finite $S$, using their binary codes ${\sum }_{n\in S}{2}^n$. For nonempty $S$ let $r=\max S$; each inclusive ordinal factor has cardinality ${\mathrm{\aleph }}_n\le {\mathrm{\aleph }}_r$, since its extra top can be sent to zero, the finite ordinals shifted by one and the other values fixed. F6 bounds the finite product by ${\mathrm{\aleph }}_r<\mu$; for empty $S$ the product has one member. Thus the countable union of these possibilities has size at most $\mu$ by A1 and F6. Every class intersected with $X_R(B)$ has size at most $\mu$, including empty classes. There are $\lambda$ possible indices, so again A1 and F6 give $\mathrm{\mid }X\mathrm{\mid }\le \lambda \cdot \mu =\lambda$.

Apply F2 to $({\alpha }_{\xi },g_{\xi }{)}_{\xi <{\omega }_1}$. The indices strictly increase, the functions are actual members of $Q$, and they strictly increase pointwise by step 1.1. Since $B\subseteq \omega \setminus \{0,1\}$, the tail $n>1$ is all of $B$. F2 gives $h(n)={\sup }_{\xi <{\omega }_1}{g}_{\xi }(n)<{\mathrm{\aleph }}_n$, $\mathrm{cf}(h(n))={\omega }_1$ at every coordinate, and $h{=}^{\ast }{f}_{\delta }$ for some $\delta <\lambda$. Thus $h\in X_R(B)$ with uniform strict bound ${\mathrm{\aleph }}_2$, and F1 gives $h\in X$. Since $g_0>b$ and $h\ge g_0$, also $h>b$. This proves pointwise cofinality and in particular nonemptiness.

Every $x\in X$ has a unique scale index $\alpha (x)$. Existence is F1; two different indices would force the same function to be eventually strictly less than itself outside finitely many coordinates, impossible on infinite $B$. Suppose $\mathrm{\mid }X\mathrm{\mid }<\lambda$. By F3–F4 choose $\beta <\lambda$ strictly above all its indices; the empty case is already excluded by step 2.1. Then $x{<}^{\ast }{f}_{\beta }$ for every $x\in X$, by F1 and each eventual equality. But step 2.1 applied to $b=f_{\beta }$ gives $h\in X$ with $f_{\beta }<h$ pointwise, contradicting $h{<}^{\ast }{f}_{\beta }$. Hence $\mathrm{\mid }X\mathrm{\mid }\ge \lambda$ by F5 and A1.

The lower bound of step 3.1 and upper bound of step 1.2 give $\mathrm{\mid }X\mathrm{\mid }=\lambda ={\mathrm{\aleph }}_{\omega +1}$. Step 2.1 establishes the asserted strict pointwise cofinality. The argument only used nonempty classes when a constructed point or an already given point provided a member. QED.