Remark 3.11.2. It is likely the case that, for some limit ordinal $\alpha $, the set of coverings $\text{Cov}(\mathcal{C})_\alpha $ satisfies the conditions of the lemma. This is after all what an application of the reflection principle would appear to give (modulo caveats as described at the end of Section 3.8 and in Remark 3.9.3).
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