A scale used in the Dowker subspace

Given the earlier $g_{\eta }$ for $\eta <\xi <{\omega }_1$, define $t_{\xi }(n)=\sup (\{1\}\cup \{g_{\eta }(n)+1:\eta <\xi \})$. By F3–F4 each countable supremum is below ${\mathrm{\aleph }}_n$; thus $t_{\xi }$ is in the strict product. The earlier indices are countable and bounded below $\lambda$ by F3–F4. F1 supplies a scale term strictly eventually above $t_{\xi }$ with index above all earlier indices, by cofinality followed by a later scale term if necessary. Let ${\alpha }_{\xi }$ be the least eligible index and put $g_{\xi }(n)=\max (t_{\xi }(n),f_{{\alpha }_{\xi }}(n))$. This is in the product, is eventually equal to $f_{{\alpha }_{\xi }}$, and is above every earlier $g_{\eta }$ pointwise. F5 now supplies the sequence by the specified rule. At stage zero, $t_0(n)=1$ and $g_0(n)=\max (1,f_{{\alpha }_0}(n))\ge 1>0$. At stage one, $t_1(n)=g_0(n)+1$ and $g_1(n)\ge g_0(n)+1$. These are the first two calculations of the instance.

Set $h(n)={\sup }_{\xi <{\omega }_1}{g}_{\xi }(n)$ and $\delta ={\sup }_{\xi <{\omega }_1}{\alpha }_{\xi }$. The sequence in step 1.1 satisfies every hypothesis of F2, and $n>1$ for every $n\in B$, so its tail is the entire coordinate set. Therefore $h(n)<{\mathrm{\aleph }}_n$, $\mathrm{cf}(h(n))={\omega }_1$, $\delta <\lambda$, and $h{=}^{\ast }{f}_{\delta }$. The cofinalities have uniform strict bound ${\mathrm{\aleph }}_2$, so $h$ is a Rudin point and F1 gives $h\in X$. The calculation $h(n)\ge g_1(n)>g_0(n)\ge 1>0$ verifies strict domination of the chosen zero bound.

For a prescribed $b$ replace $1$ in step 1.1 by $b(n)+1$. This value is below the limit cardinal ${\mathrm{\aleph }}_n$, so the same countable-supremum and least-index arguments still apply. Then $g_0(n)\ge b(n)+1>b(n)$ and the resulting supremum satisfies $h(n)\ge g_0(n)>b(n)$ for every $n$. Thus the example exhibits the actual representative construction behind pointwise cofinality, with no choice of a member from a possibly empty scale class. QED.