Difference between revisions of "Huge"

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==n huge==
 
==n huge==
  
A cardinal $\kappa$ is ''$n$-huge'' with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $M^{\kappa_n}\subset M$, where $\kappa_0=\kappa$ and $k_{i+1}=j(\kappa_i)$. Thus, the $1$-huge cardinals are precisely the huge cardinals.  
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A cardinal $\kappa$ is ''n-huge'' with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $M^{\kappa_n}\subset M$, where $\kappa_0=\kappa$ and $k_{i+1}=j(\kappa_i)$. Thus, the 1-huge cardinals are precisely the huge cardinals.  
  
  
 
==Super n huge==
 
==Super n huge==
  
A cardinal $\kappa$ is ''super-$n$-huge'' if it is $n$-huge with target $\lambda$ for arbitrarily large cardinals $\lambda$.
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A cardinal $\kappa$ is ''n-superhuge'' if it is n-huge with target $\lambda$ for arbitrarily large cardinals $\lambda$.
  
 
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Revision as of 12:22, 23 October 2017


A cardinal $\kappa$ is huge with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $j(\kappa)=\lambda$ and $M^\lambda\subset M$.


Almost huge

A cardinal $\kappa$ is almost huge with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $j(\kappa)=\lambda$ and $M^{\lt\lambda}\subset M$.

Superhuge

A cardinal $\kappa$ is superhuge if it is huge with target $\lambda$ for unboundedly many cardinals $\lambda$.

n huge

A cardinal $\kappa$ is n-huge with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $M^{\kappa_n}\subset M$, where $\kappa_0=\kappa$ and $k_{i+1}=j(\kappa_i)$. Thus, the 1-huge cardinals are precisely the huge cardinals.


Super n huge

A cardinal $\kappa$ is n-superhuge if it is n-huge with target $\lambda$ for arbitrarily large cardinals $\lambda$.


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