# Diamonds, Compactness, and Measure Sequences

## TL;DR

This paper proves the consistency of the failure of the diamond principle on certain large cardinals with reflection properties, using Radin forcing, and characterizes weak compactness in this context.

## Contribution

It introduces a novel analysis of Radin forcing to establish the failure of the diamond principle on large cardinals with reflection properties and characterizes weak compactness in Radin extensions.

## Key findings

- Failure of the diamond principle is consistent with strong reflection properties.
- Characterization of weak compactness in Radin generic extensions.
- Analysis of Radin forcing related to large cardinal properties.

## Abstract

We establish the consistency of the failure of the diamond principle on a cardinal $\kappa$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of $\kappa$ in a Radin generic extension.