# Difference between revisions of "Superstrong"

From Cantor's Attic

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A cardinal $\kappa$ is ''superstrong'' if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive class $M$, with critical point $\kappa$, such that $V_{j(\kappa)}\subset M$. | A cardinal $\kappa$ is ''superstrong'' if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive class $M$, with critical point $\kappa$, such that $V_{j(\kappa)}\subset M$. | ||

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− | + | === Relation to other large cardinal notions === | |

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+ | Every superstrong cardinal is a [[Shelah | Shelah cardinal]]. | ||

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+ | If $\kappa$ is superstrong as witnessed by $j:V \to M$, then $\kappa$ is a [[strong | strong cardinal]] in the model $M_{j(\kappa)}$. | ||

+ | However, superstrong cardinals need not be strong in $V$. | ||

+ | In fact, the least superstrong cardinal is less than the least strong cardinal, because superstrongness is a $\Sigma_2$ property (as can be seen using its characterization in terms of [[extender | extenders]]) | ||

+ | whereas every strong cardinal is $\Sigma_2$-[[reflecting]]. | ||

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## Revision as of 13:22, 21 November 2013

A cardinal $\kappa$ is *superstrong* if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive class $M$, with critical point $\kappa$, such that $V_{j(\kappa)}\subset M$.

### Relation to other large cardinal notions

Every superstrong cardinal is a Shelah cardinal.

If $\kappa$ is superstrong as witnessed by $j:V \to M$, then $\kappa$ is a strong cardinal in the model $M_{j(\kappa)}$. However, superstrong cardinals need not be strong in $V$. In fact, the least superstrong cardinal is less than the least strong cardinal, because superstrongness is a $\Sigma_2$ property (as can be seen using its characterization in terms of extenders) whereas every strong cardinal is $\Sigma_2$-reflecting.

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