If $\kappa$ is a singular cardinal, a *scale for $\kappa$* consists of an increasing sequence $\langle \kappa\_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\langle f\_\alpha : \alpha < \kappa^+ \rangle$ that is linearly ordered and dominating in the partial order of $\prod\_{i<\mathrm{cf}(\kappa)} \kappa\_i$ by $f \leq g$ iff $|\{ i : f(i) >g(i) \}| < \mathrm{cf}(\kappa)$. Shelah proved that scales exist for every singular cardinal.


A point $\alpha < \kappa^+$ is called *good* for a scale on $\kappa$ if there is an unbounded $A \subseteq \alpha$ and an $i < \mathrm{cf}(\kappa)$ such that for all $j > i$, $\langle f\_\beta(j) : \beta \in A \rangle$ is a strictly increasing sequence of ordinals.


The following seems to be folklore:


**Theorem (ZFC):** For every scale $\vec F$ for $\aleph\_\omega$, there is a club $C \subseteq \aleph\_{\omega+1}$ such that every point in $C$ of cofinality at least $\aleph\_4$ is good.


**Question 1:** Is there a proof of this in the literature? Or can one neatly construct a proof from some lemmas stated in the literature?


**Question 2:** How does this generalize to larger singular cardinals?



PCF theory and good points in scales
Jing is correct in stating that the result follows from the referenced results in the Abraham-Magidor handbook chapter.


A general theorem, which can be proven in the same way, is the following result:


**Theorem:** Suppose that $\kappa$ is a singular cardinal and $\vec{f} = \langle f\_\alpha \mid \alpha < \lambda \rangle$ is a scale on $\kappa$ (we could have $\lambda > \kappa^+$ here). Then there is a club $C \subseteq \lambda$ such that, for every regular cardinal $\mu$ with $\mathrm{cf}(\kappa) < \mu < \kappa$, and for every $\beta \in C \cap \mathrm{cof}(\mu^{+3})$, $\beta$ is good for $\vec{f}$.


**EDIT** Actually, upon further reflection, the following slightly more general result, which can again be proven in the same way, is also true:


**Theorem:** Suppose that $\kappa$ is a singular cardinal and $\vec{f} = \langle f\_\alpha \mid \alpha < \lambda \rangle$ is a scale on $\kappa$. Then there is a club $C \subseteq \lambda$ such that, for every regular cardinal $\mu$ with $\mathrm{cf}(\kappa) < \mu < \kappa$, every ordinal $\eta$ with $2 \leq \eta < \mathrm{cf}(\kappa)$, and every $\beta \in C \cap \mathrm{cof}(\mu^{+\eta+1})$, $\beta$ is good for $\vec{f}$.


