Indescribable cardinal

The Structure of Indescribability in Consistency Strength

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

Language

$Q$-indescribable cardinals are those which have the property that for every $Q$-sentence $\phi$:

$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$ By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.

One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [1]

Higher-order

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.

$\beta$-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.

Indescribable on a set

(from [1])

Language $\mathcal{L}$ has variables and quantifiers for all finite types (where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.), a name (individual constant) for each set and a name (relation symbol) for each relation on sets. (§1) TODO: complete the definition $\mathcal{L}_\in$ is the sublanguage of $\mathcal{L}$ allowing only $\in$ as a relation symbol. (above definition 1.7)

We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1)

We call $\alpha$ weakly $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1)

$R(\alpha)=\bigcup_{\beta<\alpha} \mathcal{P}(R(\beta))$ for an ordinal $\alpha$. We say that $R(\alpha)$ reflects $\varphi$ on $X$ iff $R(\alpha) \models \varphi \implies \exists_{\beta \in X \cap \alpha} R(\beta) \models \phi$. (definition 1.5)

We call $\alpha$ strongly $Q$-indescribable on $X$ iff $R(\alpha)$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.5)

We say that $L_\alpha$ reflects $\varphi$ on $X$ iff $L_\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} L_\beta \models \phi$. (definition 1.6)

We call $\alpha$ $Q$-reflecting on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}_\in$. (definition 1.7) With full $\mathcal{L}$ this would yield weak $Q$-indescribability on $X$. (above definition 1.7)

Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.

Facts

Here are some known facts about indescribability:

Weak $\Pi_2^0$-indescribability is equivalent to being uncountable and regular. (theorem 1.2)[1] Strong(definition 1.5)[1] $\Pi_2^0$-indescribability is equivalent to strong inaccessibility, $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.[2] $\Pi_1^1$-indescribability is equivalent to weak compactness. [3],[2]

The property of being a limit ($\alpha = \sup (X \cap \alpha)$) is equivalent to weak $\Pi_0^0$-indescribablity on $X$ and to weak $\Sigma_2^0$-indescribablity on $X$. Mahloness on $X$ is equivalent to weak $\Pi_2^0$-indescribablity on $X$ and to weak $\Pi_0^1$-indescribablity on $X$. Weak $\Pi_n^1$-indescribablity on $X$ is equivalent to weak $\Sigma_{n+1}^1$-indescribablity on $X$. (theorem 1.3 i-iii)[1]

If $m>2$ or $n>0$, weak $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, weak $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. ($\mathrm{Rg}$ is the class of regular cardinals.) (theorem 1.3 iv)[1]

When $Q$ is $\Pi_m^n$ or $\Sigma_m^n$ for $n>0$, an ordinal is strongly $Q$-indescribable iff it is weakly $Q$-indescribable and strongly inaccessible (therefore strong and weak $Q$-inaccessibility coincide assuming GCH.). (after definition 1.5)[1]

$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). [4]

Ineffable cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. [5]

$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. [2]

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 $m>1$.[2] 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.[2]

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.[6] If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.[7]

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.[8] Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.[8] 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.[8]

Every measurable cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. [3]

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). [3]

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 $2^\kappa\neq\kappa^+$. [9]

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 $β < \min \{α, κ^+\}$.[10]

$\mathrm{ZFC} + \mathrm{BTEE}$ (Basic Theory of Elementary Embeddings) proves that the critical point of $j$ is totally indescribable.[11]

$Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. In particular, $Π_3$-reflecting or 2-admissible ordinals can be called recursively weakly compact. (after definition 1.12)[1][12]

References

1. Richter, Wayne and Aczel, Peter. Inductive Definitions and Reflecting Properties of Admissible Ordinals. Generalized recursion theory : proceedings of the 1972 Oslo symposium, pp. 301-381, 1974. www   bibtex
2. 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
3. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
4. Rathjen, Michael. The art of ordinal analysis. , 2006. www   bibtex
5. Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www   bibtex
6. Feng, Qi. A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic 49(3):257 - 277, 1990. DOI   bibtex
7. Holy, Peter and Schlicht, Philipp. A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae 242:49-74, 2018. www   arχiv   DOI   bibtex
8. Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv   bibtex
9. Hauser, Kai. Indescribable Cardinals and Elementary Embeddings. 56(2):439 - 457, 1991. www   DOI   bibtex
10. 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
11. 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
12. Madore, David. A zoo of ordinals. , 2017. www   bibtex
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