## Tall Cardinals A cardinal $\kappa$ is **$\theta$-tall** iff there is an [](Elementary%20embedding.md) $j:V\to M$ 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](Strong.md "Strong") cardinal is tall and every [](Strongly%20compact.md) cardinal is tall, but [measurable](Measurable.md "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](Strong.md "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 %} ### Extender Characterization 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. ## Strongly Tall Cardinals A cardinal $\kappa$ is **strongly $\theta$-tall** iff there is some [measure](Filter.md "Filter") $U$ on a set $S$ witnessing $\kappas $\theta$-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](Woodin.md "Woodin") cardinal, above the consistency strength of a [strong](Strong.md "Strong") cardinal and therefore above a tall cardinal). Specifically, if $κ$ is strong and has a proper class of measurables above it and <a href="Continuum_hypothesis" class="mw-redirect" title="Continuum hypothesis">GCH</a> 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.md)). ### Ultrapower Characterization $\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and a $\kappa$-complete [ultrafilter](Filter.md "Filter") $U$ 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 $\kappa$. ### Embedding Characterization 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 $j(f)(A)=\alpha$. ### Ultrafilter Characterization $\kappa$ is strongly $\theta$-tall iff there is some set $S$, a $\kappa$-complete [ultrafilter](Filter.md "Filter") $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow V$ for each ordinal $\alpha$ such that: 1. $\kappa$ is uncountable. 2. $H_0(x)=0$ for each $x\in S$. 3. 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. 4. $\{x\in S:H_\theta(x)\in\kappa\}\in U$. That is, $H_\theta(x)\in\kappa$ almost everywhere.