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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.

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 $\theta$.

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

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