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 ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.
More generally, a cardinal number κ is called Πmn-indescribable if for every Πn proposition φ, and set A ⊆ Vκ with (Vκ+m, ∈, A) ⊧ φ there exists an α < κ with (Vα+m, ∈, A ∩ Vα) ⊧ φ.