Meeting, covering and Shelah's Revised GCH

TL;DR

This paper explores Shelah's Revised GCH Theorem's applications to diamond principles and covering numbers, proposing a generalization and proving a fragment, advancing understanding in set theory.

Contribution

It introduces a generalization of Shelah's Revised GCH Theorem and proves a fragment, expanding its applicability to covering numbers.

Findings

Revisits Shelah's Revised GCH Theorem in the context of diamond principles

Proposes a new generalization of the theorem

Proves a small fragment of the generalized theorem

Abstract

We revisit the application of Shelah's Revised GCH Theorem \cite{SheRGCH} to diamond. We also formulate a generalization of the theorem and prove a small fragment of it. Finally we consider another application of the theorem, to covering numbers of the form cov(-, -, -, $\omega$).