Compactness and Guessing Principles in the Radin Extensions

TL;DR

This paper explores the relationship between compactness and guessing principles in Radin forcing extensions, showing that certain large cardinal properties imply the diamond principle, contrasting with previous results where it fails.

Contribution

It demonstrates that in Radin extensions with weakly compact cardinals, the diamond principle can hold, providing new insights into the interaction between compactness and guessing principles.

Findings

Diamond principle holds at weakly compact cardinals in Radin extensions.

A scenario where compactness exceeds diagonal stationary reflection but diamond fails.

Improves understanding of Radin forcing effects on combinatorial principles.

Abstract

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $\kappa$, if $\kappa$ is weakly compact, then $\diamondsuit(\kappa)$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from \cite{BN19}.