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?

  • 7
    $\begingroup$ 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 Mar 26 '18 at 20:42

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 $\vec{f}$.

  • $\begingroup$ 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 Mar 28 '18 at 17:54
  • 3
    $\begingroup$ @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 Mar 28 '18 at 18:29

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.