Indescribable cardinal

In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott (1961).

A cardinal number $\kappa$ is called Πⁿ
_m-indescribable if for every Π_m proposition φ, and set A ⊆ V_κ with (V_κ+n, ∈, A) ⊧ φ there exists an α < κ with (V_α+n, ∈, A ∩ V_α) ⊧ φ. Following Lévy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. Σⁿ
_m-indescribable cardinals are defined in a similar way. The idea is that κ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties.

The cardinal number κ is called totally indescribable if it is Πⁿ
_m-indescribable for all positive integers m and n.

If $\alpha$ is an ordinal, the cardinal number $\kappa$ is called $\alpha$-indescribable if for every formula $\phi$ and every subset $U$ of $V_{\kappa }$ such that $\phi (U)$ holds in $V_{\kappa +\alpha }$ there is a some $\lambda <\kappa$ such that φ(U ∩ V_λ) holds in V_λ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if $\alpha$ is finite they are the same as $\Pi _{\omega }^{\alpha }$-indescribable ordinals. α-indescribability implies that α<κ, but there is an alternative notion of shrewd cardinals that makes sense when $\alpha \geq \kappa$: there is $\lambda <\kappa$ and $\beta$ such that φ(U ∩ V_λ) holds in V_λ+β.

Historical note

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

Equivalent conditions

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

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

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

Relationships in the large cardinal hierarchy

A cardinal is Σ¹
_n+1-indescribable if it is Π¹
_n-indescribable. The property of being Π¹
_n-indescribable is Π¹
_n+1. For m>1, the property of being Π^m
_n-indescribable is Σ^m
_n and the property of being Σ^m
_n-indescribable is Π^m
_n. Thus, for m>1, every cardinal that is either Π^m
_n+1-indescribable or Σ^m
_n+1-indescribable is both Π^m
_n-indescribable and Σ^m
_n-indescribable and the set of such cardinals below it is stationary. The consistency strength is Σ^m
_n-indescribable cardinals is below that of Π^m
_n-indescribable, but for m>1 it is consistent with ZFC that the least Σ^m
_n-indescribable exists and is above the least Π^m
_n-indescribable cardinal (this is proved from consistency of ZFC with Π^m
_n-indescribable cardinal and a Σ^m
_n-indescribable cardinal above it).

Measurable cardinals are Π²
₁-indescribable, but the smallest measurable cardinal is not Σ²
₁-indescribable. However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal.

Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for Π^m
_n and Σ^m
_n indescribability.

References

• 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.