Approximating diamond principles on products at an inaccessible cardinal

TL;DR

This paper introduces approximating diamond principles at inaccessible cardinals, uses them to develop new methods for negating the diamond principle at large cardinals, and explores their implications at weakly compact cardinals.

Contribution

It defines approximating diamond principles, demonstrates their use in negating the diamond principle without altering cofinalities, and establishes their necessity at weakly compact cardinals.

Findings

Approximating diamond principles hold at weakly compact cardinals.

New forcing method for negating diamond at large cardinals without changing cofinalities.

Obstacles to failing the diamond principle at weakly compact cardinals identified.

Abstract

We isolate \emph{the approximating diamond principles}, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik's overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal $\theta$ without changing cofinalities or adding fast clubs to $\theta$. In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined with the fact that in all known models where the diamond principle fails the approximating diamond principles also fail at an inaccessible cardinal, exhibits essential combinatorial obstacles to make the diamond principle fail at a weakly compact cardinal.