Vopěnka's principle

From Cantor's Attic
Revision as of 02:07, 16 January 2012 by Andrewbt (Talk | contribs) (A start to the page: the definition & place in the hierarchy.)

Jump to: navigation, search


Vopěnka's principle is a large cardinal axiom at the upper end of the large cardinal hierarchy that is particularly notable for its applications to category theory. In a set theoretic setting, the most common definition is the following:

For any language $\mathcal{L}$ and any proper class $C$ of $\mathcal{L}$-structures, there are distinct structures $M, N\in C$ and an elementary embedding $j:M\to N$.

For example, taking $\mathcal{L}$ to be the language with one unary and one binary predicate, we can consider for any ordinal $\eta$ the class of structures $\langle V_{\alpha+\eta},\{\alpha\},\in\rangle$, and conclude from Vopěnka's principle that a cardinal that is at least $\eta$-extendible exists. In fact if Vopěnka's principle holds then there are a proper class of extendible cardinals; bounding the strength of the axiom from above, we have that if $\kappa$ is almost huge, then $V_\kappa$ satisfies Vopěnka's principle.


    This article is a stub. Please help us to improve Cantor's Attic by adding information.

Vopěnka cardinal

    This article is a stub. Please help us to improve Cantor's Attic by adding information.