Difference between revisions of "Highjump"
(Created page with "''Highjump'' cardinals are a certain kind of large cardinals. A cardinal $\kappa$ is ''highjump'' if it is the critical point of an elementary embedding $j:V\to M$ s...") 
(No difference)

Latest revision as of 03:53, 13 September 2017
Highjump cardinals are a certain kind of large cardinals. A cardinal $\kappa$ is highjump 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.
Contents
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 $j$.
A cardinal $\kappa$ is highjump 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 highjump embedding, and a normal fine ultrafilter on some $P_\kappa\lambda$ generating an ultrapower embedding that is highjump is highjump ultrafilter (or highjump measure).
$\kappa$ is called almost highjump 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 highjump 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 highjump cardinals which allows to have one embedding per $f:\kappa\to\kappa$ rather than a single embedding for all such $f$s.
$\kappa$ is highjump order $\eta$ (resp. almost highjump order $\eta$) if there exists a strictly increasing sequence of ordinals $\{\theta_\alpha\alpha<\eta\}$ such that for all $\alpha<\eta$, there exists a highjump embedding (resp. almost highjump embedding) with critical point $\kappa$ and clearance $\theta_\alpha$. $\kappa$ is super highjump (resp. super almost highjump) if there are highjump embeddings (resp. almost highjump embeddings) with arbtirarily large clearance (i.e. it is "(almost) highjump order $Ord$").
A highjump cardinal $\kappa$ has unbounded excess closure if for some clearance $\theta$, for all cardinals $\lambda\geq\theta$, there is a highjump measure on $P_\kappa\lambda$ generating an embedding with clearance $\theta$.
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 $j(f)(\kappa)=\theta$.
 $\kappa^{+}\leq cf(\theta)\leq 2^\kappa$ (see cofinality)
 $\beth^M_\theta=\theta$
 $M_\theta\prec M_{j(\kappa)}$
 $M_\theta\models ZFC$
The following holds when $\kappa$ is almost highjump, and holds in both $V$ and $M$:
 $\theta^\kappa=\theta$
 $\theta$ is singular.
 $V_\theta\prec M_{j(\kappa)}$
 $V_\theta\models ZFC$
The following statements also holds:
 Suppose there is a almost highjump 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 highjump cardinal is almost highjump, and has order $\theta$; in fact, in the models $V_\theta$, $V_\kappa$ and $M_{j(\kappa)}$ there are many super almost highjump cardinals.
 The existence of a highjump cardinal with order $\eta$ implies that for every $\gamma<\eta$, there exists a model in which that cardinal is highjump with order $\gamma$. The same statement holds for almost highjump cardinals.
 The existence of a highjump cardinal with unbounded excess closure is equiconsistent with the existence of a cardinal $\kappa$ such that for all sufficiently large $\lambda$, there exists a highjump measure on $P_\kappa\lambda$.
 Suppose $\kappa$ is almost huge; then in the model $V_\kappa$ there are many cardinals that are highjump with unbounded excess closure.
 Suppose that there exists a pair of cardinals ($\kappa$, $\theta$) such that there is a highjump 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 highjump in the model $V_\theta$, and the cardinal $\kappa$ has highjump order $\theta$ in $V$. Furthermore, there are many superhighjump cardinals in the models $V_\kappa$, $V_\theta$, and $M_{j(\kappa)}$.
 The least highjump cardinal is not $\Sigma_2$reflecting. In particular, it is not supercompact and not even strong. The same is true for the least almosthuge cardinal, the least almosthighjump cardinal, and the least Shelahforsupercompactness cardinal.
Name origin
Read More
 Norman Lewis Perlmutter, The large cardinals between supercompact and almosthuge [1]
This article is a stub. Please help us to improve Cantor's Attic by adding information.