# Difference between revisions of "Tall"

From Cantor's Attic

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{{DISPLAYTITLE: Tall cardinal}} | {{DISPLAYTITLE: Tall cardinal}} | ||

− | A cardinal $\kappa$ is ''tall'' if for every ordinal $\theta$ there is an elementary embedding $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. Every [[strong]] cardinal is tall and every [[strongly compact]] cardinal is tall, but [[measurable]] cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a [[strong]] cardinal. Any tall cardinal $\kappa$ can be made indestructible by a variety of forcing notions, | + | A cardinal $\kappa$ is ''tall'' if for every ordinal $\theta$ there is an [[elementary embedding]] $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. Every [[strong]] cardinal is tall and every [[strongly compact]] cardinal is tall, but [[measurable]] cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a [[strong]] cardinal. Any tall cardinal $\kappa$ can be made indestructible by a variety of forcing notions, |

including forcing that pumps up the value of $2^\kappa$ as high as desired. See <cite>Hamkins2009:TallCardinals</cite>. | including forcing that pumps up the value of $2^\kappa$ as high as desired. See <cite>Hamkins2009:TallCardinals</cite>. | ||

## Revision as of 23:57, 2 October 2017

A cardinal $\kappa$ is *tall* if for every ordinal $\theta$ there is an elementary embedding $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal $\kappa$ can be made indestructible by a variety of forcing notions,
including forcing that pumps up the value of $2^\kappa$ as high as desired. See [1].

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