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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?

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    $\begingroup$ Sharon and Viale's paper (SOME CONSEQUENCES OF REFLECTIONON THE APPROACHABILITY IDEAL) says Theorem2.13and Lemmas 2.12 and 2.19 of the handbook chapter by Abraham and Magidor gives the result. $\endgroup$
    – Jing Zhang
    Commented Mar 26, 2018 at 20:42

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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 $\vec{f}$.

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  • $\begingroup$ So the way I read it, you can look to Theorem 2.21 in the same reference to conclude that almost all points of cofinality at least $\omega_4$ on a scale for $\aleph_\omega$ have an exact upper bound. But to say that point is good, we would want this eub to have cofinality $\omega_4$ almost everywhere. It doesn't look like we get that. What am I missing? $\endgroup$
    – Monroe Eskew
    Commented Mar 28, 2018 at 17:54
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    $\begingroup$ @MonroeEskew Theorem 2.21 gives you an eub for points of cofinality $\geq \omega_4$, but, crucially, it in fact gives you an eub $h$ such that $\mathrm{cf}(h(n)) \geq \aleph_1$ for all $n < \omega$. Then an argument using the fact that there are only finitely many cardinals between $\aleph_1$ and the cofinality of the point you're looking at gives you that actually $\mathrm{cf}(h(n))$ must equal that cofinality for almost all $n$. A similar argument will work in the general case. Arguments like this can be found in Lemmas 7 and 8 of Magidor and Shelah's "When does almost free imply free?" $\endgroup$
    – Chris Lambie-Hanson
    Commented Mar 28, 2018 at 18:29
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Shelah has also considered this question in his paper [Sh:1008]. The published version indicates that he investigated this from scratch rather than starting with the Sharon-Viale observation on the Abraham-Magidor Handbook article. I haven't read the paper in detail, so I don't know how much he is able to prove, but I do know he obtains results on the extent $I[\lambda]$, as well as some results on the extent of good points for scales.

[Sh:1008] Shelah, S., Non-reflection of the bad set for $\check{I}_\theta[\lambda]$ and pcf, Acta Math. Hung. 141, No. 1-2, 11-35 (2013). ZBL1324.03014.

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