Representability and Compactness for Pseudopowers

Abstract

We prove a compactness theorem for pseudopower operations of the form $p{p}_{\Gamma (\mu ,\sigma )}(\mu )$ where ${\aleph }_0<\sigma =cf(\sigma )\le cf(\mu )$. 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.

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