Shrewd cardinal

In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by (Rathjen 1995), 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 ⊆ V_κ with (V_κ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (V_α+λ', ∈, A ∩ V_α) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ[1]^{(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.[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.

More generally, a cardinal number κ is called λ-Π_m-shrewd if for every Π_m proposition φ, and set A ⊆ V_κ with (V_κ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (V_α+λ', ∈, A ∩ V_α) ⊧ φ.[1]^{(Definition 4.1)} Π_m 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-Π_m-shrewd cardinals is the same thing as a Π_mⁿ-indescribable cardinal.

If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in κ.[1]^{(Lemma 4.6)} Rathjen does not state how shrewd cardinals compare to unfoldable cardinals, however.

λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (V_α+λ, ∈, A ∩ V_α), 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.