For an ordinal λ, a cardinal number κ is called λ-shrewd if for every proposition φ using a predicate symbol and with one free variable, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ(κ) there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ(α).
[1](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.
[2] λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (Vα+λ, ∈, A ∩ Vα), making it impossible for a cardinal κ to be κ-indescribable.
Also, the monotonicity property is lost: a λ-indescribable cardinal may fail to be α-indescribable for some ordinal α < λ.