An extension of Shelah’s trichotomy theorem

Abstract

Shelah (Algorithms Comb 14:420–459, 1997) 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.