A cardinal $\kappa$ is *indescribable* if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$: $\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$ Likewise for $\Sigma_{m}^n$-indescribable cardinals. Here are some other equivalent definitions: - A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$: $\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$ - A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$: $\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$ In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [shrewd](Shrewd.md "Shrewd") cardinals, an extension of indescribable cardinals. ## Variants *Totally indescribable* cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. *$Q$-indescribable* cardinals are those which have the property that for every $Q$-sentence $\phi$: $\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$ *$\beta$-indescribable* cardinals are those which have the property that for every first order sentence $\phi$: $\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$ There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable. ## Facts Here are some known facts about indescribability: $\Pi_2^0$-indescribability is equivalent to [](Inaccessible.md), $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.{% cite Kanamori2009 %} $\Pi_1^1$-indescribability is equivalent to [[weakly compact]]. {% cite Jech2003 Kanamori2009 %} $\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). {% cite Rathjen2006 %} [Ineffable](Ineffable.md "Ineffable") cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. {% cite Jensen1969 %} $\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. {% cite Kanamori2009 %} If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.{% cite Kanamori2009 %} If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.{% cite Kanamori2009 %} If $\kappa$ is $Π_n$-[Ramsey](Ramsey.md "Ramsey"), then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.{% cite Feng1990 %} If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.{% cite Holy2018 %} Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.{% cite Nielsen2018 %} Every lt;ω$-Ramsey cardinal is $∆^2_0$-indescribable.{% cite Nielsen2018 %} Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.{% cite Nielsen2018 %} Every [measurable](Measurable.md "Measurable") cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. {% cite Jech2003 %} Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [ZFC](ZFC.md "ZFC") is totally indescribable in $M$. (For example, <a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">rank-into-rank</a> cardinals, <a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^{\#}lt;/a> cardinals, and <a href="Zero_dagger" class="mw-redirect" title="Zero dagger">$0^{\dagger}lt;/a> cardinals). {% cite Jech2003 %} If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. {% cite Hauser1991 %} Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[Ramsey](Ramsey.md "Ramsey"), then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.{% cite Sharpe2011 %}