[Paper Review] A class of strong diamond principles

Joel David Hamkins|ArXiv.org|Nov 26, 2002

Advanced Topology and Set Theory10 references20 citations

TL;DR

This paper introduces and investigates a class of strengthened diamond principles—Laver diamond principles—generalizing the classical diamond principle $ abla_ ext{κ}$ to weaker large cardinal notions such as measurable, weakly compact, and unfoldable cardinals. It establishes that these principles, which allow embeddings to realize arbitrary sets via a Laver function, can hold or fail independently of the large cardinal properties, offering new combinatorial characterizations and forcing possibilities.

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.

Motivation & Objective

• To generalize the Laver function concept—originally defined for supercompact cardinals—to weaker large cardinal notions such as measurable, weakly compact, and unfoldable cardinals.
• To formulate and analyze new strong diamond principles, denoted $\mathop{\hbox{\textasciitilde}}^{\star}_{\kappa}$, that strengthen $\Diamond_\kappa$ by requiring anticipation of subsets on sets of measure one or via elementary embeddings.
• To investigate the consistency and independence of these Laver diamond principles from classical diamond principles and from one another, especially in forcing extensions.
• To determine whether these principles can be forced to fail while preserving the large cardinal properties of $\kappa$, particularly in the context of measurable and $\theta$-supercompact cardinals.
• To clarify the logical relationship between the Laver diamond principle and $\Diamond_\kappa(\text{Reg})$, especially for weakly compact cardinals.

Proposed method

• Define the Laver diamond principle $\mathop{\hbox{\textasciitilde}}^{\text{meas}}_{\kappa}$ for measurable cardinals as the existence of a function $\ell: \kappa \to V_\kappa$ such that for every $A \in H(\kappa^+)$, there exists an elementary embedding $j: V \to M$ with critical point $\kappa$ and $j(\ell)(\kappa) = A$.
• Extend this framework to other large cardinals—weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable, and strongly compact—by adapting the embedding-based anticipation condition to each cardinal's characteristic ultrapower or embedding properties.
• Use recursive construction of partial functions $\ell \upharpoonright \gamma$ to build Laver functions, ensuring that at each stage $\gamma$, the value $\ell(\gamma)$ is chosen to preserve the potential for future embedding realization.
• Apply forcing techniques, including class forcing and forcing extensions preserving large cardinals, to demonstrate that $\mathop{\hbox{\textasciitilde}}^{\star}_{\kappa}$ can fail while preserving the large cardinal property of $\kappa$.
• Leverage results from Hauser (1992) and Dzamonja–Hamkins to force $\neg\Diamond_\kappa(\text{Reg})$ while preserving weakly compact or $\Pi^m_n$-indescribable cardinals, thereby showing that the Laver diamond principle can fail in such models.
• Analyze the relationship between $\mathop{\hbox{\textasciitilde}}^{\star}_{\kappa}$ and $\Diamond_\kappa(\text{Reg})$, particularly in the context of weakly compact cardinals, using model-theoretic and forcing arguments to assess equivalence or separation.

Experimental results

Research questions

• RQ1 Is the Laver diamond principle $\mathop{\hbox{\textasciitilde}}^{\text{wc}}_{\kappa}$ equivalent to $\Diamond_\kappa(\text{Reg})$ when $\kappa$ is weakly compact?
• RQ2 Can one force the failure of $\mathop{\hbox{\textasciitilde}}^{\text{meas}}_{\kappa}$ while preserving the measurability of $\kappa$ using a less destructive forcing than class forcing?
• RQ3 Is it relatively consistent that $\kappa$ is $\theta$-supercompact with $\kappa < \theta$, but $\mathop{\hbox{\textasciitilde}}^{\theta\text{-sc}}_{\kappa}$ fails?
• RQ4 Can the number of normal fine measures on $P_\kappa\theta$ be limited to fewer than $2^\theta$ while preserving $\theta$-supercompactness, thereby implying failure of the $\theta$-supercompact Laver diamond principle?
• RQ5 Do the Laver diamond principles for strongly unfoldable and Ramsey cardinals also admit independent failure via forcing, even when the large cardinal properties are preserved?

Key findings

• For measurable cardinals, the Laver diamond principle $\mathop{\hbox{\textasciitilde}}^{\text{meas}}_{\kappa}$ is equivalent to the existence of a function $\ell: \kappa \to V_\kappa$ such that for every $A \in H(\kappa^+)$, there is an elementary embedding $j: V \to M$ with $j(\ell)(\kappa) = A$.
• The classical diamond principle $\Diamond_\kappa(\text{Reg})$ must hold for measurable cardinals, but $\mathop{\hbox{\textasciitilde}}^{\text{meas}}_{\kappa}$ can fail, showing that the Laver diamond principle is strictly stronger than $\Diamond_\kappa(\text{Reg})$ in the measurable context.
• Using Hauser’s result, it is relatively consistent that $\kappa$ is weakly compact but $\mathop{\hbox{\textasciitilde}}^{\text{wc}}_{\kappa}$ fails, as $\neg\Diamond_\kappa(\text{Reg})$ can be forced while preserving weak compactness.
• For strongly unfoldable cardinals, Dzamonja and Hamkins have shown that $\mathop{\hbox{\textasciitilde}}^{\text{sunf}}_{\kappa}$ can also fail in a forcing extension preserving the large cardinal property.
• The existence of a Laver function for a large cardinal is not guaranteed by the cardinal's existence alone; it is a new combinatorial principle that can hold or fail independently.
• The failure of $\mathop{\hbox{\textasciitilde}}^{\theta\text{-sc}}_{\kappa}$ for $\theta$-supercompact $\kappa$ is implied by having fewer than $2^\theta$ normal fine measures on $P_\kappa\theta$, which is a consistent possibility under certain forcing conditions.