# Tall cardinals

• J. D. Hamkins, “Tall cardinals,” Math.~Logic Q., vol. 55, iss. 1, p. 68–86, 2009.
[Bibtex]
@ARTICLE{Hamkins2009:TallCardinals,
AUTHOR = {Hamkins, Joel D.},
TITLE = {Tall cardinals},
JOURNAL = {Math.~Logic Q.},
FJOURNAL = {Mathematical Logic Quarterly},
VOLUME = {55},
YEAR = {2009},
NUMBER = {1},
PAGES = {68--86},
ISSN = {0942-5616},
MRCLASS = {03E55 (03E35)},
MRNUMBER = {2489293 (2010g:03083)},
MRREVIEWER = {Carlos A.~Di Prisco},
DOI = {10.1002/malq.200710084},
URL = {http://wp.me/p5M0LV-3y},
file = F,
}

A cardinal $\kappa$ is tall if for every ordinal $\theta$ there is an embedding $j:V\to M$ with critical point $\kappa$ such that $j(\kappa)\gt\theta$ 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.