Shrewd cardinal
# 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 $λ^{(Definition4.1)}$ (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 < $λ.^{(Corollary4.3)}$ Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of $Π^1_2-comprehension.$ 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 ∩ $V_α)$ ⊧ $φ.^{(Definition4.1)}$ $Π_m$ 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 $Π_m^n-indescribable$ cardinal.

If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in $κ.^{(Lemma4.6)}$ 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 $(V_{α+λ},$ ∈, 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.

