A class of strong diamond principles

TL;DR

This paper explores strengthened diamond principles associated with large cardinals, generalizing the Laver function concept to various large cardinal notions and examining their consistency and implications.

Contribution

It introduces and analyzes Laver diamond principles for multiple large cardinal notions, extending the classical diamond principle in new and meaningful ways.

Findings

Laver diamond principles can hold or fail depending on the large cardinal context.

These principles generalize the Laver function concept beyond supercompact cardinals.

The paper establishes connections between diamond principles and large cardinal embeddings.

Abstract

In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset of kappa not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function l:kappa-->V_kappa such that for any A in H(kappa+) there is an embedding j:V-->M having critical point kappa with j(l)(kappa)=A. This and similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardinals for these weaker large cardinal notions. The resulting Laver diamond principles can hold or fail in a variety of interesting ways.