Nearly $\theta$-supercompact cardinals

• Formal definition
• Characterizations of near $\theta$-supercompactness
• Nearly strongly compact

The near $\theta$-supercompactness hierarchy of cardinals was introduced by Jason Schanker in (Schanker, 2012) and (Schanker, 2011). The hierarchy stratifies the $\theta$-supercompactness hierarchy in the sense that every $\theta$-supercompact cardinal is nearly $\theta$-supercompact, and every nearly $2^{\theta^{ {<}\kappa}}$-supercompact cardinal $\kappa$ is $\theta$-supercompact. However, these cardinals can be very different. For example, relative to the existence of a supercompact cardinal $\kappa$ with an inaccessible cardinal $\theta$ above it, we can force to destroy $\kappa$’s measurability while still retaining its near $\theta$-supercompactness and the weak inaccessibility of $\theta$. Yet, if $\theta^{ {<}\kappa} = \theta$ and $\kappa$ is $\theta$-supercompact, we can also force to preserve $\kappa$’s $\theta$-supercompactness while destroying any potential near $\theta^+$-supercompactness without collapsing cardinals below $\theta^{++}$. Assuming that $\theta^{ {<}\kappa} = \theta$, nearly $\theta$-supercompact cardinals $\kappa$ exhibit a hybrid of weak compactness and supercompactness in that the witnessing embeddings are between $\text{ZFC}^-$ ($\text{ZFC}$ minus the powerset axiom) models of size $\theta$ but are generated by “partially normal” fine filters on $\mathcal{P}_{\kappa}(\theta$). Weakly compact cardinals $\kappa$ are nearly $\kappa$-supercompact.

Formal definition

A cardinal $\kappa$ is nearly $\theta$-supercompact if and only if for every $A\subseteq\theta$, there exists a transitive $M \vDash ZFC^{-}$ closed under ${<}\kappa$ sequences with $A, \kappa, \theta \in M$, a transitive $N$, and an elementary embedding $j: M \rightarrow N$ with critical point $\kappa$ such that $j(\kappa) > \theta$ and $j’’\theta \in N$. A cardinal is nearly supercompact if it is nearly $\theta$-supercompact for all $\theta$.

Characterizations of near $\theta$-supercompactness

If $\theta^{ {<}\kappa} = \theta$, then the following are equivalent characterizations for the near $\theta$-supercompactness of $\kappa$:

Embedding 
For every ${<}\kappa$-closed transitive set $M$ of size $\theta$ with $\theta \in M$, there exists a transitive $N$ and an elementary embedding $j: M \rightarrow N$ with critical point $\kappa$ such that $j(\kappa) > \theta$ and $j’’\theta \in N$.

Normal Embedding 

Normal ZFC Embedding 

Normal Fine Filter 
For every family of subsets $\mathcal{A} \subset \mathcal{P}_\kappa(\theta)$ of size at most $\theta$ and every collection $\mathcal{F}$ of at most $\theta$ many functions from $\mathcal{P}_{\kappa}(\theta)$ into $\theta$, there exists a $\kappa$-complete fine filter $F$ on $\mathcal{P}_{\kappa}(\theta)$, which is $\mathcal{F}$-normal in the sense that for every $f \in \mathcal{F}$ that’s regressive on some set in $F$, there exists $\alpha_f < \theta$ for which $\{\sigma \in \mathcal{P}_{\kappa}(\theta)| f(\sigma) = \alpha_f\} \in F$.

Hauser Embedding 

Nearly strongly compact

References

1. Schanker, J. A. (2012). Partial near supercompactness. Ann. Pure Appl. Logic. https://doi.org/10.1016/j.apal.2012.08.001
2. Schanker, J. A. (2011). Weakly measurable cardinals and partial near supercompactness [PhD thesis]. CUNY Graduate Center.

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