*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%20embedding.md) $j:V\to M$ 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](Huge.md "Huge") cardinal. ## Definition 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 [](Filter.md) on some $\mathcal{P}_\kappa(\lambda)$ generating an [](Ultrapower.md) 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 lt;\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 $f:\kappa\to\kappa$, $M^{j(f)(\kappa)}\subseteq M$. <a href="Shelah" class="mw-redirect" title="Shelah">Shelah for supercompactness</a> 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](Superstrong.md "Superstrong"). ## Properties 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 <a href="Cofinality" class="mw-redirect" title="Cofinality">cofinality</a>) 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 <a href="Singular" class="mw-redirect" title="Singular">singular</a>. Moreover, $V_\theta\prec M_{j(\kappa)}$ and $V_\theta$ satisfies ZFC. The following statements also holds: - Suppose there is a almost high-jump cardinal. Then there are many cardinals below it that are Shelah for supercompactness. Also, in the model $V_\kappa$ there are many supercompact cardinals. - Every high-jump cardinal is almost high-jump, and has order $\theta$; in fact, in the models $V_\theta$, $V_\kappa$ and $M_{j(\kappa)}$ there are many super almost high-jump cardinals. - The existence of a high-jump cardinal with order $\eta$ implies that for every $\gamma<\eta$, there exists a model in which that cardinal is high-jump with order $\gamma$. The same statement holds for almost high-jump cardinals. - The existence of a high-jump cardinal with unbounded excess closure is equiconsistent with the existence of a cardinal $\kappa$ such that for all sufficiently large $\lambda$, there exists a high-jump measure on $\mathcal{P}_\kappa(\lambda$). - Suppose $\kappa$ is [](Huge.md); then in the model $V_\kappa$ there are many cardinals that are high-jump with unbounded excess closure. - Suppose that there exists a pair of cardinals ($\kappa$, $\theta$) such that there is a high-jump embedding $j:V\to M$ with critical point $\kappa$ and clearance $\theta$ and such that $M^{2^\theta}\subseteq M$. Then the cardinal $\kappa$ is super high-jump in the model $V_\theta$, and the cardinal $\kappa$ has high-jump order $\theta$ in $V$. Furthermore, there are many super high-jump cardinals in the models $V_\kappa$, $V_\theta$, and $M_{j(\kappa)}$. - The least high-jump cardinal is not $\Sigma_2$-reflecting. In particular, it is not supercompact and not even strong. The same is true for the least [](Huge.md) cardinal, the least [superstrong](Superstrong.md "Superstrong") cardinal, the least almost-high-jump cardinal, and the least Shelah-for-supercompactness cardinal. ## Name origin ## Read More - Norman Lewis Perlmutter, *The large cardinals between supercompact and almost-huge* <a href="http://boolesrings.org/perlmutter/files/2013/07/HighJumpForJournal.pdf" class="external autonumber">[1]</a> This article is a stub. Please help us to improve Cantor's Attic by adding information.