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High-jump cardinals are a certain kind of large cardinals. A cardinal $\kappa$ is high-jump if it is the critical point of an elementary embedding $j:V\to M$ such that $M$ is closed under sequences of length $\text{sup}\{j(f)(\kappa)|f:\kappa\to\kappa\}$. This closure condition is a weakening of the definition of a huge cardinal.


Let $j:M\to N$ be a (nontrivial) elementary embedding. The clearance of $j$ is the ordinal $\text{sup}\{j(f)(\kappa)$ $|$ $f:\kappa\to\kappa,f\in M\}$ where $\kappa$ is the critical point of $j$.

A cardinal $\kappa$ is high-jump if there exists $j:V\to M$ with critical point $\kappa$ and clearance $\theta$ such that $M^\theta\subseteq M$, i.e. $M$ contains all sequences of elements of $M$ of length $\theta$. $j$ is called a high-jump embedding, and a normal fine ultrafilter on some $\mathcal{P}_\kappa(\lambda)$ generating an ultrapower embedding that is high-jump is a high-jump ultrafilter (or high-jump measure).

$\kappa$ is called almost high-jump if $M$ is closed under sequences of length $<\theta$ instead, i.e. $M^\lambda\subseteq M$ for all $\lambda<\theta$. $j$ is then an almost high-jump embedding. This means that for all $f:\kappa\to\kappa$, $M^{j(f)(\kappa)}\subseteq M$. Shelah for supercompactness cardinals are a natural weakening of almost high-jump cardinals which allows to have one embedding per $f:\kappa\to\kappa$ rather than a single embedding for all such $f$s.

$\kappa$ is high-jump order $\eta$ (resp. almost high-jump order $\eta$) if there exists a strictly increasing sequence of ordinals $\{\theta_\alpha:\alpha<\eta\}$ such that for all $\alpha<\eta$, there exists a high-jump embedding (resp. almost high-jump embedding) with critical point $\kappa$ and clearance $\theta_\alpha$. $\kappa$ is super high-jump (resp. super almost high-jump) if there are high-jump embeddings (resp. almost high-jump embeddings) with arbtirarily large clearance (i.e. it is “(almost) high-jump order Ord”).

A high-jump cardinal $\kappa$ has unbounded excess closure if for some clearance $\theta$, for all cardinals $\lambda\geq\theta$, there is a high-jump measure on $\mathcal{P}_\kappa(\lambda)$ generating an ultrapower embedding with clearance $\theta$.

The dual notion high-jump-for-strongness, where the closure condition $M^\theta\subseteq M$ is weakened to $V_\theta\subseteq M$, turns out to be equivalent to superstrongness.


Let $j:V\to M$ a nontrivial elementary embedding with critical point $\kappa$ and clearance $\theta$. Then there is no $f:\kappa\to\kappa$ such that $j(f)(\kappa)=\theta$. Also, $\kappa^{+}\leq cf(\theta)\leq 2^\kappa$ (see cofinality) and $\beth^M_\theta=\theta$. Moreover, $M_\theta\prec M_{j(\kappa)}$ and $M_\theta$ satisfies ZFC where $M_\theta=M\cap V_\theta$.

When $\kappa$ is almost high-jump, in both $V$ and $M$, $\theta^\kappa=\theta$, also $\theta$ is singular. Moreover, $V_\theta\prec M_{j(\kappa)}$ and $V_\theta$ satisfies ZFC.

The following statements also holds:

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