# Difference between revisions of "Vopenka"

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− | {{DISPLAYTITLE: Vopěnka cardinal}} | + | {{DISPLAYTITLE: Vopěnka's principle}} |

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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: | ||

+ | <blockquote> | ||

+ | 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 $j:M\to N$. | ||

+ | </blockquote> | ||

+ | 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 [[huge#Almost huge|almost huge]], then $V_\kappa$ satisfies Vopěnka's principle. | ||

{{stub}} | {{stub}} | ||

− | + | ==Vopěnka cardinal== | |

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− | == Vopěnka | + | |

− | + | ||

{{stub}} | {{stub}} |

## Revision as of 02:07, 16 January 2012

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.

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## Vopěnka cardinal

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