An Extension of Shelah's Trichotomy Theorem

Abstract

In Sh506, Shelah develops the theory of ${\mathrm{pcf}}_I(A)$ without the assumption that $|A|<\min (A)$, going so far as to get generators for every $\lambda \in {\mathrm{pcf}}_I(A)$ under some assumptions on $I$. Our main theorem is that we can also generalize Shelah's trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for ${\mathrm{pcf}}_I(A)$ which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.

arXiv comment17 pages, updated the journal reference. This article appears as in The Archive for Mathematical Logic. The final authenticated version is available online at: https://doi.org/10.1007/s00153-018-0631-6

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