Difference between revisions of "Vopenka"

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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.

As stated above and from the point of view of ZFC, this is actually an axiom schema, as we quantify over proper classes, which from a purely ZFC perspective means definable proper classes. One alternative is to view Vopěnka's principle as an axiom in a class theory, such as von Neumann-Gödel-Bernays. Another is to consider a _Vopěnka cardinal_, that is, a cardinal $\kappa$ that is inaccessible and such that $V_\kappa$ satisfies Vopěnka's principle when "proper class" is taken to mean "subset of $V_\kappa$ of cardinality $\kappa$. These three alternatives are, in the order listed, strictly increasing in strength (see http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably).

Equivalent statements

The schema form of Vopěnka's principle is equivalent to the existence of a proper class of $C^{(n)}$-extendible cardinals for every $n$; indeed there is a level-by-level stratification of Vopěnka's principle, with Vopěnka's principle for a $\Sigma_{n+2}$-definable class corresponds to the existence of a $C^{(n)}$-extendible cardinal greater than the ranks of the parameters. [1]

Vopěnka cardinal

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  1. Bagaria, Joan and Casacuberta, Carles and Mathias, A R D and Rosický, Jiří. Definable orthogonality classes in accessible categories are small. Journal of the European Mathematical Society 17(3):549--589. arχiv   bibtex
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