# An Extension of Shelah's Trichotomy Theorem

## TL;DR

This paper extends Shelah's pcf theory and trichotomy theorem to broader settings without size restrictions, providing new proofs and discussing challenges for further generalizations.

## Contribution

It generalizes Shelah's trichotomy theorem to the setting where |A| is not less than min(A), and offers a modern proof of generator existence in pcf theory.

## Key findings

- Generalization of Shelah's trichotomy theorem
- New proof of generator existence in pcf theory
- Discussion of obstacles to further generalizations

## 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.