Indescribable cardinal

Originally, a cardinal κ was called Q-indescribable if for every Q-formula ${\displaystyle \phi }$ and relation ${\displaystyle A}$, if ${\displaystyle (\kappa ,<,A)\vDash \phi }$ then there exists an ${\displaystyle \alpha <\kappa }$ such that ${\displaystyle (\alpha ,\in ,A\upharpoonright \alpha )\vDash \phi }$.[4][5] Using this definition, ${\displaystyle \kappa }$ is ${\displaystyle \Pi _{0}^{1}}$-indescribable iff ${\displaystyle \kappa }$ is regular and greater than ${\displaystyle \aleph _{0}}$.[5]^p.207 The cardinals ${\displaystyle \kappa }$ satisfying the above version based on the cumulative hierarchy were called strongly Q-indescribable.[6]

A cardinal is ${\displaystyle \Sigma _{n+1}^{1}}$-indescribable iff it is ${\displaystyle \Pi _{n}^{1}}$-indescribable.[7] A cardinal is inaccessible if and only if it is ${\displaystyle \Pi _{n}^{0}}$-indescribable for all positive integers ${\displaystyle n}$, equivalently iff it is ${\displaystyle \Pi _{2}^{0}}$-indescribable, equivalently if it is ${\displaystyle \Sigma _{1}^{1}}$-indescribable.

${\displaystyle \Pi _{1}^{1}}$-indescribable cardinals are the same as weakly compact cardinals.

The indescribability condition is equivalent to ${\displaystyle V_{\kappa }}$ satisfying the reflection principle (which is provable in ZFC), but extended by allowing higher-order formulae with a second-order free variable.[7]

For cardinals ${\displaystyle \kappa <\theta }$, say that an elementary embedding ${\displaystyle j:M\to H(\theta )}$ a small embedding if ${\displaystyle M}$ is transitive and ${\displaystyle j({\textrm {crit}}(j))=\kappa }$. For any natural number ${\displaystyle 1\leq n}$, ${\displaystyle \kappa }$ is ${\displaystyle \Pi _{n}^{1}}$-indescribable iff there is an ${\displaystyle \alpha >\kappa }$ such that for all ${\displaystyle \theta >\alpha }$ there is a small embedding ${\displaystyle j:M\to H_{\theta }}$ such that ${\displaystyle H({\textrm {crit}}(j)^{+})^{M}\prec _{\Sigma _{n}}H({\textrm {crit}}(j)^{+})}$.[8]^{, Corollary 4.3}

If V=L, then for a natural number n>0, an uncountable cardinal is Π¹
_n-indescribable iff it's (n+1)-stationary.[9]

For a class ${\displaystyle X}$ of ordinals and a ${\displaystyle \Gamma }$-indescribable cardinal ${\displaystyle \kappa }$, ${\displaystyle X}$ is said to be enforced at ${\displaystyle \alpha }$ (by some formula ${\displaystyle \phi }$ of ${\displaystyle \Gamma }$) if there is a ${\displaystyle \Gamma }$-formula ${\displaystyle \phi }$ and an ${\displaystyle A\subseteq V_{\kappa }}$ such that ${\displaystyle (V_{\kappa },\in ,A)\vDash \phi }$, but for no ${\displaystyle \beta <\alpha }$ with ${\displaystyle \beta \notin X}$ does ${\displaystyle (V_{\beta },\in ,A\cap V_{\beta })\vDash \phi }$ hold.[1]^p.277 This gives a tool to show necessary properties of indescribable cardinals.

The property of ${\displaystyle \kappa }$ being ${\displaystyle \Pi _{n}^{1}}$-indescribable is ${\displaystyle \Pi _{n+1}^{1}}$ over ${\displaystyle V_{\kappa }}$, i.e. there is a ${\displaystyle \Pi _{n+1}^{1}}$ sentence that ${\displaystyle V_{\kappa }}$ satisfies iff ${\displaystyle \kappa }$ is ${\displaystyle \Pi _{n}^{1}}$-indescribable.[10] For ${\displaystyle m>1}$, the property of being ${\displaystyle \Pi _{n}^{m}}$-indescribable is ${\displaystyle \Sigma _{n}^{m}}$ and the property of being ${\displaystyle \Sigma _{n}^{m}}$-indescribable is ${\displaystyle \Pi _{n}^{m}}$.[10] Thus, for ${\displaystyle m>1}$, every cardinal that is either ${\displaystyle \Pi _{n+1}^{m}}$-indescribable or ${\displaystyle \Sigma _{n+1}^{m}}$-indescribable is both ${\displaystyle \Pi _{n}^{m}}$-indescribable and ${\displaystyle \Sigma _{n}^{m}}$-indescribable and the set of such cardinals below it is stationary. The consistency strength of ${\displaystyle \Sigma _{n}^{m}}$-indescribable cardinals is below that of ${\displaystyle \Pi _{n}^{m}}$-indescribable, but for ${\displaystyle m>1}$ it is consistent with ZFC that the least ${\displaystyle \Sigma _{n}^{m}}$-indescribable exists and is above the least ${\displaystyle \Pi _{n}^{m}}$-indescribable cardinal (this is proved from consistency of ZFC with ${\displaystyle \Pi _{n}^{m}}$-indescribable cardinal and a ${\displaystyle \Sigma _{n}^{m}}$-indescribable cardinal above it).

Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for ${\displaystyle \Pi _{n}^{m}}$- and ${\displaystyle \Sigma _{n}^{m}}$-indescribability.

Measurable cardinals are ${\displaystyle \Pi _{1}^{2}}$-indescribable, but the smallest measurable cardinal is not ${\displaystyle \Sigma _{1}^{2}}$-indescribable. However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal.

For ${\displaystyle n\geq 1}$, ZFC+"there is a ${\displaystyle \Sigma _{n}^{1}}$-indescribable cardinal" is equiconsistent with ZFC+"there is a ${\displaystyle \Sigma _{n}^{1}}$-indescribable cardinal ${\displaystyle \kappa }$ such that ${\displaystyle 2^{\kappa }>\kappa ^{+}}$", i.e. "GCH fails at a ${\displaystyle \Sigma _{n}^{1}}$-indescribable cardinal".[7]

• Hanf, W. P.; Scott, D. S. (1961), "Classifying inaccessible cardinals", Notices of the American Mathematical Society, 8: 445, ISSN 0002-9920
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. doi:10.1007/978-3-540-88867-3_2. ISBN 3-540-00384-3.