# Representability and Compactness for Pseudopowers

## TL;DR

This paper establishes a compactness theorem for pseudopower operations, linking it to Shelah's cov vs. pp Theorem, and explores the implications of non-compactness for pcf theory and the structure of regular cardinals.

## Contribution

It proves a new compactness theorem for pseudopower operations and connects it to Shelah's cov vs. pp Theorem, revealing consequences for pcf theory.

## Key findings

- Proves a compactness theorem for pseudopower operations.
- Shows failure of compactness implies existence of a set with an inaccessible accumulation point in pcf theory.
- Links compactness results to Shelah's cov vs. pp Theorem.

## Abstract

We prove a compactness theorem for pseudopower operations of the form $pp_{\Gamma(\mu,\sigma)}(\mu)$ where $\aleph_0<\sigma=cf(\sigma)\leq cf(\mu)$. Our main tool is a result that has Shelah's cov vs. 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 $pcf(A)$ has an inaccessible accumulation point.