Indescribable cardinal

• Variants
• Facts

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.(Kanamori, 2009) $\Pi_1^1$-indescribability is equivalent to weak compactness. (Jech, 2003; Kanamori, 2009)

$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). (Rathjen, 2006)

Ineffable cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. (Jensen & Kunen, 1969)

$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. (Kanamori, 2009)

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 $m>1$.(Kanamori, 2009) 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.(Kanamori, 2009)

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.(Feng, 1990) If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.(Holy & Schlicht, 2018)

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.(Nielsen & Welch, 2018) Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.(Nielsen & Welch, 2018) 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.(Nielsen & Welch, 2018)

Every measurable cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. (Jech, 2003)

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). (Jech, 2003)

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 $2^\kappa\neq\kappa^+$. (Hauser, 1991)

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 $β < \min \{α, κ^+\}$.(Sharpe & Welch, 2011)

References

1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
2. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf
3. Rathjen, M. (2006). The art of ordinal analysis. http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf
4. Jensen, R., & Kunen, K. (1969). Some combinatorial properties of L and V. http://www.mathematik.hu-berlin.de/ raesch/org/jensen.html
5. Feng, Q. (1990). A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic, 49(3), 257–277. https://doi.org/10.1016/0168-0072(90)90028-Z
6. Holy, P., & Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae, 242, 49–74. https://doi.org/10.4064/fm396-9-2017
7. Nielsen, D. S., & Welch, P. (2018). Games and Ramsey-like cardinals.
8. Hauser, K. (1991). Indescribable Cardinals and Elementary Embeddings. 56(2), 439–457. https://doi.org/10.2307/2274692
9. Sharpe, I., & Welch, P. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann. Pure Appl. Logic, 162(11), 863–902. https://doi.org/10.1016/j.apal.2011.04.002

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