Shrewd

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(Almost all information from [1])

Shrewd cardinals are a generalisation of indescribable cardinals. They are called shrewd because they are bigger in size than many large cardinals with much greater consistency strength (for all notions of large cardinal which do not make reference to the totality of all ordinals).

Definitions

Let $κ$ be a cardinal, $η>0$ be an ordinal, and $\mathcal{A}$ be a class.

$κ$ is $η$-shrewd iff for all $X ⊆ V_κ$ and for every formula $\phi(x_1, x_2)$, if $V_{κ+η} \models \phi(X, κ)$, then $\exists_{0 < κ_0, η_0 < κ} V_{κ_0+η_0} \models \phi(X ∩ V_{κ_0}, κ_0)$.

$κ$ is shrewd iff $κ$ is $η$-shrewd for every $η > 0$.

$κ$ is $\mathcal{A}$-$η$-shrewd iff for all $X ⊆ V_κ$ and for every formula $\phi(x_1, x_2)$, if $\langle V_{κ+η}, \mathcal{A} ∩ V_{κ+η} \rangle \models \phi(X, κ)$, then $\exists_{0 < κ_0, η_0 < κ} \langle V_{κ_0+η_0}, \mathcal{A} ∩ V_{κ_0+η_0} \rangle \models \phi(X ∩ V_{κ_0}, κ_0)$.

$κ$ is $\mathcal{A}$-shrewd iff $κ$ is $\mathcal{A}$-$η$-shrewd for every $η > 0$.

One can also use a collection of formulae $\mathcal{F}$ and make $\phi$ an $\mathcal{F}$-formula to define $η$-$\mathcal{F}$-shrewd and $\mathcal{A}$-$η$-$\mathcal{F}$-shrewd cardinals.

Properties

  • If $κ$ is $\mathcal{A}$-$δ$-shrewd and $0 < η < δ$, then $κ$ is $\mathcal{A}$-$η$-shrewd. This is a difference between the properties of shrewdness and indescribability.
  • For subtle $\pi$,
    • for every class $\mathcal{A}$, in every club $B ⊆ π$ there is $κ$ such that $\langle V_\pi, \mathcal{A} ∩ V_\pi \rangle \models \text{“$κ$ is $\mathcal{A}$-shrewd .”}$. (The set of cardinals $κ$ below $\pi$ that are $\mathcal{A}$-shrewd in $V_\pi$ is stationary.)
    • there is an $\eta$-shrewd cardinal below $\pi$ for all $\eta < \pi$.
  • Shrewdness implies $\Pi_2$-reflection (cardinal).
  • The shrewd cardinals and the strongly unfoldable cardinals coincide. [1]

References

  1. Rathjen, Michael. The art of ordinal analysis. , 2006. www   bibtex
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