A cardinal \(\kappa\) is \(\Pi^n_m\)-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.
\(\Pi^1_1\)-indescribable cardinals are the same as weakly compact cardinals.
Ordinals, ordinal analysis and set theory
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