About: Indescribable cardinal   Sponge Permalink

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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.

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  • Indescribable cardinal
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  • 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.
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abstract
  • 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.
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