Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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A cardinal $\kappa$ is **$\theta$-tall** iff there is an elementary
embedding
$j:V\to M$ into a transitive class $M$ with critical point $\kappa$
such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. $\kappa$
is **tall** iff it is $\theta$-tall for every $\theta$; i.e.
$j(\kappa)$ can be made arbitrarily large. 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
(Hamkins, 2009)

If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-extender $E$ such that, if $M\cong Ult_E$ is the ultrapower of $V$ by $E$, $M^\kappa\subset M$. Similarly, $\kappa$ is tall iff for any $\lambda$ there exists some $(\kappa,\lambda)$-extender such that $M^\kappa\subset M$ where $M$ is as above.

A cardinal $\kappa$ is **strongly $\theta$-tall** iff there is some
measure
$U$ on a set $S$ witnessing $\kappa$’s $\theta$-tallness in the
ultrapower of $V$ by $U$. More precisely, the ultrapower embedding
$j:V\prec M$ has critical point $\kappa$, $M^\kappa\subset M$, and
$j(\kappa)>\theta$. $\kappa$ is **strongly tall** iff it is
strongly $\theta$-tall for every $\theta$.

The existence of a strongly tall cardinal is equiconsistent to the existence of a strong cardinal with a proper class of measurables above it (below the consistency strength of a Woodin cardinal, above the consistency strength of a strong cardinal and therefore above a tall cardinal). Specifically, if $κ$ is strong and has a proper class of measurables above it and GCH holds, then in a forcing extension of $V$, $κ$ is strongly tall. On the other hand, if $κ$ is strongly tall and there is no inner model with two strong cardinals, then $κ$ is strong in $K$ and has a proper class of measurables above it in $K$ ($K$ being the core model).

$\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and a $\kappa$-complete ultrafilter $U$ on $S$ such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)>\theta$. That is, there is an ultrapower of an ultrafilter which witnesses the $\gamma$-tallness of $\kappa$.

If $\theta\geq\kappa$, then $\kappa$ is strongly $\theta$-tall iff $\kappa$ is the critical point of some $j:V\prec M$ for which there is a set $S$ and an $A\in j(S)$ such that for any $\alpha\leq\theta$, there is a function $f:S\rightarrow\kappa$ with $j(f)(A)=\alpha$.

$\kappa$ is strongly $\theta$-tall iff there is some set $S$, a $\kappa$-complete ultrafilter $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow V$ for each ordinal $\alpha$ such that:

- $\kappa$ is uncountable.
- $H_0(x)=0$ for each $x\in S$.
- For each $\alpha$ and each $f:S\rightarrow V$, $\{x\in S:f(x)\in H_\alpha(x)\}\in U$ iff there is some $\beta<\alpha$ such that $\{x\in S:f(x)=H_\beta(x)\}\in U$. That is, $f(x)\in H_\alpha(x)$ almost everywhere iff there is some $\beta<\alpha$ such that $f(x)=H_\beta(x)$ almost everywhere.
- $\{x\in S:H_\theta(x)\in\kappa\}\in U$. That is, $H_\theta(x)\in\kappa$ almost everywhere.

- Hamkins, J. D. (2009). Tall cardinals.
*MLQ Math. Log. Q.*,*55*(1), 68–86. https://doi.org/10.1002/malq.200710084