Indescribable Cardinal

In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe 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 κ 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. 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 α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of V_κ such that φ(U) holds in V_κ+α there is a some λ<κ such that φ(U ∩ V_λ) holds in V_λ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Πα
ω-indescribable ordinals. α-indescribability implies that α<κ, but there is an alternative notion that makes sense when α≥κ: there is λ<κ and β such that φ(U ∩ V_λ) holds in V_λ+β.

Π1
1-indescribable cardinals are the same as weakly compact cardinals.

A cardinal is inaccessible if and only if it is Π0
n-indescribable for all positive integers n, equivalently iff it is Π0
2-indescribable, equivalently iff it is Σ1
1-indescribable. A cardinal is Σ1
n+1-indescribable iff it is Π1
n-indescribable. The property of being Π1
n-indescribable is Π1
n+1. For m>1, the property of being Πm
n-indescribable is Σm
n and the property of being Σm
n-indescribable is Πm
n. Thus, for m>1, every cardinal that is either Πm
n+1-indescribable or Σm
n+1-indescribable is both Πm
n-indescribable and Σm
n-indescribable and the set of such cardinals below it is stationary. The consistency strength is Σm
n-indescribable cardinals is below that of Πm
n-indescribable, but for m>1 it is consistent with ZFC that the least Σm
n-indescribable exists and is above the least Πm
n-indescribable cardinal (this is proved from consistency of ZFC with Πm
n-indescribable cardinal and a Σm
n-indescribable cardinal above it).

Measurable cardinals are Π2
1-indescribable, but the smallest measurable cardinal is not Σ2
1-indescribable. However 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 Πm
n and Σm
n indescribability.