Shrewd cardinal


In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by., extending the definition of indescribable cardinals.
A cardinal number κ is called λ-shrewd if for every proposition φ, and set A ⊆ Vκ with ⊧ φ there exists an α, λ' < κ with ⊧ φ. It is called shrewd if it is λ-shrewd for every λ.
This definition extends the concept of indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for any ordinal μ < λ. 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 ⊧ φ there exists an α, λ' < κ with ⊧ φ.
Here 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. 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, 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.