Indescribable cardinal
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 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 strong inaccessibility, $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.[1] $\Pi_1^1$-indescribability is equivalent to weak compactness. [2],[1]
$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). [3]
Ineffable cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. [4]
$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. [1]
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 $m>1$.[1] 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.[1]
If $\kappa$ is $Π_n$-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.[5] If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.[6]
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.[7] Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.[7] 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.[7]
Every measurable cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. [2]
Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of ZFC is totally indescribable in $M$. (For example, rank-into-rank cardinals, $0^{\#}$ cardinals, and $0^{\dagger}$ cardinals). [2]
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 $2^\kappa\neq\kappa^+$. [8]
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, then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.[9]
$\mathrm{ZFC} + \mathrm{BTEE}$ (Basic Theory of Elementary Embeddings) proves that the critical point of $j$ is totally indescribable.[10]
References
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex
- Rathjen, Michael. The art of ordinal analysis. , 2006. www bibtex
- Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex
- Feng, Qi. A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic 49(3):257 - 277, 1990. DOI bibtex
- Holy, Peter and Schlicht, Philipp. A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae 242:49-74, 2018. www arχiv DOI bibtex
- Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv bibtex
- Hauser, Kai. Indescribable Cardinals and Elementary Embeddings. 56(2):439 - 457, 1991. www DOI bibtex
- Sharpe, Ian and Welch, Philip. Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann Pure Appl Logic 162(11):863--902, 2011. www DOI MR bibtex
- Corazza, Paul. The spectrum of elementary embeddings $j : V \to V$. Annals of Pure and Applied Logic 139(1--3):327-399, May, 2006. DOI bibtex