Concept

Shrewd cardinal

Summary
In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by , extending the definition of indescribable cardinals. For an ordinal λ, a cardinal number κ is called λ-shrewd if for every proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ(Definition 4.1) (including λ > κ). This definition extends the concept of indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for any ordinal μ < λ.(Corollary 4.3) Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of Π12-comprehension. It is essentially the nonrecursive analog to the stability property for admissible ordinals. More generally, a cardinal number κ is called λ-Πm-shrewd if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ.(Definition
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