High-jump
(from [1])
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.
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 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.
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 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:
- 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 almost huge; 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 almost huge cardinal, the least superstrong cardinal, the least almost-high-jump cardinal, and the least Shelah-for-supercompactness cardinal.
Name origin
References
- Perlmutter, Norman. The large cardinals between supercompact and almost-huge. , 2010. www arχiv bibtex
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