cantors-attic

Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

Quick navigation
The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar

Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine

Shrewd

(All information from (Rathjen, 2006))

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

Definitions

$κ$ — cardinal, $η>0$ — ordinal, $\mathcal{A}$ — 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

References

  1. Rathjen, M. (2006). The art of ordinal analysis. http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf
Main library