# Tall cardinal

From Cantor's Attic

Revision as of 11:06, 3 January 2012 by Jdh (Talk | contribs) (Created page with "{{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 critic...")

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.

This article is a stub. Please help us to improve Cantor's Attic by adding information.