Good Scales and Non-Compactness of Squares

Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $κ$ is a singular strong limit of uncountable cofinality, all scales on $κ$ are good, and $\square^_δ$ holds for all $δ<κ$, then $\square_κ^$ holds. In this paper we will present a strongly contrasting result for $\aleph_ω$. We construct a model in which $\square_{\aleph_n}$ holds for all $n<ω$, all scales on $\aleph_ω$ are good, but in which $\square_{\aleph_ω}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ω+1})]^{\aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results.

💡 Research Summary

The paper investigates the compactness of square principles at singular cardinals, focusing on the case of ℵ ω, a singular strong limit of countable cofinality. Earlier work by Cummings, Foreman, and Magidor showed that if a singular strong limit κ of uncountable cofinality has all scales good and the weak square □*δ holds for every δ < κ, then the weak square □*κ also holds. Lev22 later proved a similar compactness result for such κ. The present work provides a striking contrast: it constructs a model in which every ordinary square □ℵₙ (for n < ω) holds, all scales on ℵ ω are good, yet the weak square □*ℵ ω fails, and many countable‑sized elementary substructures of H(ℵ ω + 1) are not sup‑internally approachable.

The construction assumes the consistency of a κ^{+ ω + 1}‑supercompact cardinal. Starting from such a large‑cardinal background, the authors first force a “good‑scale” extension that adds a family of functions forming a scale on ℵ ω with the property that every point of cofinality > cf ℵ ω is a good point (i.e., there are exact upper bounds with the appropriate cofinalities). This is done using a relatively mild forcing that does not disturb the large‑cardinal structure.

The second, crucial step is a variant of Namba forcing, denoted L. Conditions in L are finite‑height trees whose nodes at level n carry values from ℵ_{d(n)} where d:ω→ω{0,1} is chosen so that each finite value appears infinitely often and the first occurrence of a value m is preceded only by smaller values. The tree has a unique stem, and every node extending the stem splits into a stationary set of successors of cofinality ω₁. This design guarantees two technical properties: (1) L forces ℵ ω to become countably cofinal while preserving ℵ ω + 1 as a regular cardinal of cofinality at least ℵ₁, and (2) L preserves stationary subsets of ω₁ because the splitting sets are large enough to support the standard stationary‑preservation argument (a game‑theoretic approach similar to Cummings‑Magidor and Krueger).

Two key lemmas about L are proved. Theorem 2.1 (an approximation lemma) states that in the two‑step iteration L * U, where U is a countably closed forcing, any function with domain ω₁ that is cofinal in a cardinal of cofinality ≥ ℵ ω + 1 must already appear in the ground model on an initial segment. This shows that L does not add new ω₁‑sequences of ordinals beyond what is forced by the subsequent collapse, a crucial ingredient for preserving H(θ) for large θ. Proposition 2.2 (a decision lemma) shows that any name for an ordinal below ω₁ can be decided below a suitable extension of a given condition, using a fusion sequence and a gluing argument to eliminate “bad” nodes where the decision fails. Together these lemmas give L the “approximation + covering” behavior needed to control square sequences.

With the good‑scale forcing and L in place, the authors analyze the interaction. The exact upper bounds added by L have cofinality ω₁, which by Fact 1.7 ensures that the scale remains good after forcing. However, the club sequence generated by L destroys the weak square □*ℵ ω: the club guessing required by □ℵ ω cannot be met because the Namba forcing adds a new cofinal ω‑sequence through ℵ ω that cannot be threaded by a coherent □‑sequence. Consequently, □ℵₙ holds for each finite n (the ordinary squares are obtained from the good points of the scale as in Cummings‑Foreman‑Magidor), but □*ℵ ω fails.

Finally, the paper studies internal approachability. Using the same forcing, the authors produce stationarily many elementary substructures N ≺ H(ℵ ω + 1) of size ℵ₁ that are not sup‑internally approachable at ℵ ω + 1. This demonstrates a failure of a weak form of internal approachability that had previously been shown consistent with the presence of good scales.

The paper also presents variations, such as a model where □(ℵ ω + 1, ℵ₁) fails, showing tension between good scales and stronger square principles that would follow from the Proper Forcing Axiom. The results illuminate the delicate balance between PCF theory (good scales), combinatorial principles (square and its weak variants), and forcing constructions (Namba forcing), and they provide a new counterexample to the belief that good scales automatically yield weak square at singular successors. The techniques developed—especially the refined Namba forcing with stationary splitting and the approximation lemmas—are likely to be useful in further investigations of singular cardinal combinatorics and related compactness phenomena.