Abstract
We prove a compactness theorem for pseudopower operations of the form \({{\,\mathrm{pp}\,}}_{\Gamma (\mu ,\sigma )}(\mu )\) where \(\aleph _0<\sigma ={{\,\mathrm{cf}\,}}(\sigma )\le {{\,\mathrm{cf}\,}}(\mu )\). Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which \({{\,\mathrm{pcf}\,}}(A)\) has an inaccessible accumulation point.
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Notes
This follows easily from work of Shelah; see Corollary 6.3 in this paper.
The original proof of the cov vs. pp Theorem (Theorem 5.4 of Chapter III of [8]) is perfectly fine, although it has a weaker conclusion and the methods do not seem to give the applications we derive here. That proof was also written before the existence of generators was proved, so it can also be simplified quite a bit.
This is standard pcf theory: see Theorem 4.4 of [1], and use the fact that \(\lambda =\max {{\,\mathrm{pcf}\,}}(B_\lambda [A])\).
This appears in published form at the end of the book [8], but the document – known as [E:12] – has been updated many times, and is available on Shelah’s Archive.
See also the last section of [5].
To get a positive answer to Question 1, just take \(\sigma ={{\,\mathrm{cf}\,}}(\mu )\) in Theorem 6.8.
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Eisworth, T. Representability and compactness for pseudopowers. Arch. Math. Logic 61, 55–80 (2022). https://doi.org/10.1007/s00153-021-00780-9
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DOI: https://doi.org/10.1007/s00153-021-00780-9