Totally indescribable cardinal

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In mathematics, a totally indescribable cardinal is a certain kind of large cardinal number.

Formally, a cardinal number κ is called totally indescribable iff for every natural number n, 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.

The idea is that κ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic 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.

More generally, 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.

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