Difference between revisions of "Vopenka"

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{{DISPLAYTITLE: Vopěnka cardinal}}
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{{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. 
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In a set theoretic setting, the most common definition is the following:
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<blockquote>
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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 $j:M\to N$.
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</blockquote>
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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
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$\langle V_{\alpha+\eta},\{\alpha\},\in\rangle$, and conclude from Vopěnka's principle that a cardinal that is at least $\eta$-[[extendible]] exists.
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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
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if $\kappa$ is [[huge#Almost huge|almost huge]], then $V_\kappa$ satisfies Vopěnka's principle.
  
 
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==Vopěnka cardinal==
 
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== Vopěnka's principle ==  
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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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