Difference between revisions of "Vopenka"

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(Added not necessarily weakly compact and external links section.)
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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.
 
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
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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 $\kappa$.
 
$\kappa$ 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).
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These three alternatives are, in the order listed, strictly increasing in strength [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably].
  
 
==Equivalent statements==
 
==Equivalent statements==
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<CITE>BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses</CITE>
 
<CITE>BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses</CITE>
  
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==Other points to note==
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Whilst Vopěnka cardinals are very strong in terms of consistency strength, a Vopěnka cardinal need not even be [[weakly compact]].  Indeed, the definition of a Vopěnka cardinal is a $\Pi^1_1$ statement
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over $V_\kappa$, and $\Pi^1_1$ indescribability is one of the equivalent definitions of weak compactness.  Thus, the least weakly compact Vopěnka cardinal must have (many) other Vopěnka cardinals less than it.
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==External links==
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  * [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably Math Overflow question and answer about formalisations]
  
 
{{References}}
 
{{References}}

Revision as of 01:25, 4 February 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.

Formalisations

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 [1].

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]

Other points to note

Whilst Vopěnka cardinals are very strong in terms of consistency strength, a Vopěnka cardinal need not even be weakly compact. Indeed, the definition of a Vopěnka cardinal is a $\Pi^1_1$ statement over $V_\kappa$, and $\Pi^1_1$ indescribability is one of the equivalent definitions of weak compactness. Thus, the least weakly compact Vopěnka cardinal must have (many) other Vopěnka cardinals less than it.

External links

  * Math Overflow question and answer about formalisations

References

  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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