Difference between revisions of "High-jump"

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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 $sup\{j(f)(\kappa)|f:\kappa\to\kappa\}$. This closure condition is a weakening of the definition of a [[huge]] cardinal.
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(from <cite>Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge</cite>)
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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.
  
 
== Definition ==
 
== Definition ==
  
Let $j:M\to N$ be a (nontrivial) elementary embedding. The ''clearance'' of $j$ is the ordinal $sup\{j(f)(\kappa)|f:\kappa\to\kappa\}$ where $\kappa$ is the critical point of $j$.
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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 $M^\theta\subset M$, i.e. $M$ contains all sequences of elements of $M$ of length $\theta$. $j$ is called a ''high-jump embedding'', and a [[filter|normal fine ultrafilter]] on some $P_\kappa\lambda$ generating an [[ultrapower|ultrapower embedding]] that is high-jump is ''high-jump ultrafilter'' (or ''high-jump measure'').
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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|normal fine ultrafilter]] on some $\mathcal{P}_\kappa(\lambda)$ generating an [[ultrapower|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. $M^\lambda\subset M$ for all $\lambda<\theta$. $j$ is then an ''almost high-jump'' embedding. This means that for all $f:\kappa\to\kappa$, $M^{j(f)(\kappa)}\subset M$. [[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.
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$\kappa$ 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|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 $Ord$").
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$\kappa$ 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 $P_\kappa\lambda$ generating an embedding with clearance $\theta$.
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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 $\theta$.
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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 [[superstrong|superstrongness]].
  
 
== Properties ==
 
== 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 $j(f)(\kappa)=\theta$.
 
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 $M_\theta\models ZFC$ where $M_\theta=M\cap V_\theta$.
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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 $V_\theta\models ZFC$
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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:
 
The following statements also holds:
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* 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 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 $P_\kappa\lambda$.  
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* 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|almost huge]]; then in the model $V_\kappa$ there are many cardinals that are high-jump with unbounded excess closure.
 
* Suppose $\kappa$ is [[huge|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 $M^{2^\theta}\subset M$. 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 $M_{j(\kappa)}$.
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* 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|almost huge]] cardinal, the least [[superstrong]] cardinal, the least almost-high-jump cardinal, and the least Shelah-for-supercompactness cardinal.
 
* 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|almost huge]] cardinal, the least [[superstrong]] cardinal, the least almost-high-jump cardinal, and the least Shelah-for-supercompactness cardinal.
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== Name origin ==
 
== Name origin ==
  
== Read More ==
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{{References}}
* Norman Lewis Perlmutter, ''The large cardinals between supercompact and almost-huge'' [http://boolesrings.org/perlmutter/files/2013/07/HighJumpForJournal.pdf]
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{{stub}}
 
{{stub}}
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[[Category:Large cardinal axioms]]
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[[Category:Critical points]]

Latest revision as of 11:06, 24 August 2021

(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

  1. Perlmutter, Norman. The large cardinals between supercompact and almost-huge. , 2010. www   arχiv   bibtex
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