Shrewd cardinal
Introduction 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' α+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 (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' α+λ', ∈, A ∩ Vα) ⊧ φ. (Definition 4.1) Πm is one of the levels of the Lévy hierarchy, in short one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.For finite ''n'', an ''n''-Πm-shrewd cardinals is the same thing as a Πmn-indescribable cardinal.If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in κ. (Lemma 4.6) A cardinal is strongly unfoldable iff it is shrewd.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.
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