Difference between revisions of "Strongly compact"

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The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an
 
The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an
 
uncountable cardinal $\kappa$ is ''strongly compact'' if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$.  
 
uncountable cardinal $\kappa$ is ''strongly compact'' if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$.  
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The strongly compact embedding characterizations are closely related to that of [[supercompact]] cardinals, which are characterized by embeddings with a high degree of closure: $\kappa$ is $\theta$-[[supercompact]] if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta<j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement to insist only that $M$ contains a small cover for any subset of size $\theta$, or even just a small cover of the set $j''\theta$ itself, we arrive at the $\theta$-strongly compact cardinals. It follows that every $\theta$-[[supercompact]] cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Furthermore, since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\of M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.
  
 
=== Strong compactness characterization ===
 
=== Strong compactness characterization ===
  
An uncountable cardinal $\kappa$ is ''strongly compact'' if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. This is called the ''$\kappa$-compactness'' property for the logic $L_{\kappa,\kappa}$, which allows for infinitary conjunctions and disjunctions of size less than $\kappa$, and blocks of quantifiers $\exists\vec x$ or $\forall\vec x$ quantifying over fewer than $\kappa$ many variables at once.
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An uncountable cardinal $\kappa$ is ''strongly compact'' if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha<\delta}\varphi_\alpha$ of any size $\delta<\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\<x_\alpha\st\alpha<\delta>$ of size less than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics. A theory is ''$\kappa$-satisfiable'' if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be ''strongly compact'' if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the ''$\kappa$-compactness property}''). The cardinal $\kappa$ is [[weakly compact]], in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.
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=== Strong compactness embedding characterization ===
 
=== Strong compactness embedding characterization ===
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A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an elementary embedding $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$.  
 
A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an elementary embedding $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$.  
  
:; Fine measure characterization An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete fine
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=== Fine measure characterization ===
     measure on $P_\kappa\theta$.
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An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete fine
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     measure on $P_\kappa\theta$. The notation $P_\kappa\theta$ means $\{\sigma\subset\theta\mid |\sigma|<\kappa\}$. A filter $F$ on $P_\kappa\theta$, measuring subsets of $P_\kappa\theta$, is ''fine'' if for every $\alpha<\theta$ the set $\{\sigma\in P_\kappa\theta\mid \alpha\in\sigma\}$ is in $F$.
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=== Cover property characterization ===
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A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding      $j:V\to M$, with critical point $\kappa$, that exhibits the ''$\theta$-strong compactness cover property'', meaning that for every $t\of M$ of size $\theta$ there is $s\in M$ with $t\of s$ and $|s|^M<j(\kappa)$.
  
:; Strong compactness covering embedding characterization : A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding      $j:V\to M$, with critical point $\kappa$, that exhibits the ''$\theta$-strong compactness cover property'', meaning that for every $t\of M$ of size $\theta$ there is $s\in M$ with $t\of s$ and $|s|^M<j(\kappa)$.
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=== Filter extension characterization ===
  
:; Strong compactness filter extension characterization : An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete
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An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete
 
     filter of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set.
 
     filter of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set.
  
:; Discontinuous ultrapower characterization : A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda<j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$.
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=== Discontinuous ultrapower characterization ===
  
:; Discontinuous embeddings characterization : A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with    $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda<j(\lambda)$.
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A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda<j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. In other words, the embedding is discontinuous at all such $\lambda$.  
  
:; Ketonen characterization : An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^\ltkappa$.
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=== Discontinuous embedding characterization ===
  
:; Regular ultrafilter characterization : An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set.  
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A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with    $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda<j(\lambda)$.
  
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=== Ketonen characterization ===
  
The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha<\delta}\varphi_\alpha$ of any size $\delta<\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\<x_\alpha\st\alpha<\delta>$ of size less than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics. A theory is ''$\kappa$-satisfiable'' if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be ''strongly compact'' if every
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An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^\ltkappa$. An ultrafilter $\mu$ on a cardinal $\lambda$ is ''uniform'' if all final segments $[\beta,\lambda)= \{\alpha<\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality.  
$\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the ''$\kappa$-compactness property}''). The cardinal $\kappa$ is [[weakly compact]], in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.
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The strongly compact embedding characterization is closely related to that of supercompactness, which generalizes it. Earlier in this book, we saw that supercompactness is characterized by the existence of embeddings exhibiting a high degree of closure. For example, $\kappa$ is $\theta$-supercompact if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta<j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement somewhat, to insist instead only that $M$ contains a small cover for any subset of size $\theta$, we arrive at the notion of $\theta$-strongly compact cardinals, which will ramify the strong compactness concept into a proper hierarchy. Specifically, I define that a cardinal $\kappa$
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=== Regular ultrafilter characterization ===
is {\df $\theta$-strongly compact} if there is an embedding $j:V\to M$ with critical point $\kappa$ such that for every $t\subset M$ of size at most $\theta$ there is $s\in M$ with $t\of s$ and $|s|^M<j(\kappa)$. I will refer to this property as the $\theta$-strongly compact ''covering property''. Clearly, any $\theta$-supercompact cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\of M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.
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The notation $P_\kappa\theta$ means $\{\sigma\subset\theta\mid |\sigma|<\kappa\}$. A filter $F$ on $P_\kappa\theta$, measuring subsets of $P_\kappa\theta$, is ''fine'' if for every $\alpha<\theta$ the set $\{\sigma\in P_\kappa\theta\mid \alpha\in\sigma\}$ is in $F$. An embedding $j:V\to M$ is ''discontinuous'' at $\lambda$ if $\sup j''\lambda<j(\lambda)$. The critical point of an embedding, for example, is its smallest discontinuity point. An ultrafilter $\mu$ on a cardinal $\lambda$ is ''uniform'' if all final segments $[\beta,\lambda)= \{\alpha<\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality. An ultrafilter $\mu$ is ''$(\kappa,\theta)$-regular'' if it is $\kappa$-complete and there is a family $\set{X_\alpha\st\alpha<\theta}\of \mu$ such that $\intersect_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.
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An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set. An ultrafilter $\mu$ is ''$(\kappa,\theta)$-regular'' if it is $\kappa$-complete and there is a family $\set{X_\alpha\st\alpha<\theta}\of \mu$ such that $\intersect_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.

Revision as of 09:51, 3 January 2012


The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an uncountable cardinal $\kappa$ is strongly compact if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$.

The strongly compact embedding characterizations are closely related to that of supercompact cardinals, which are characterized by embeddings with a high degree of closure: $\kappa$ is $\theta$-supercompact if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta<j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement to insist only that $M$ contains a small cover for any subset of size $\theta$, or even just a small cover of the set $j''\theta$ itself, we arrive at the $\theta$-strongly compact cardinals. It follows that every $\theta$-supercompact cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Furthermore, since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\of M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.

Strong compactness characterization

An uncountable cardinal $\kappa$ is strongly compact if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha<\delta}\varphi_\alpha$ of any size $\delta<\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\<x_\alpha\st\alpha<\delta>$ of size less than $\kappa$. A theory in such a language is satisfiable if it has a model under the natural semantics. A theory is $\kappa$-satisfiable if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be strongly compact if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the $\kappa$-compactness property}). The cardinal $\kappa$ is weakly compact, in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.


Strong compactness embedding characterization

A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an elementary embedding $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$.

Fine measure characterization

An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete fine

    measure on $P_\kappa\theta$. The notation $P_\kappa\theta$ means $\{\sigma\subset\theta\mid |\sigma|<\kappa\}$. A filter $F$ on $P_\kappa\theta$, measuring subsets of $P_\kappa\theta$, is fine if for every $\alpha<\theta$ the set $\{\sigma\in P_\kappa\theta\mid \alpha\in\sigma\}$ is in $F$.

Cover property characterization

A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$, with critical point $\kappa$, that exhibits the $\theta$-strong compactness cover property, meaning that for every $t\of M$ of size $\theta$ there is $s\in M$ with $t\of s$ and $|s|^M<j(\kappa)$.

Filter extension characterization

An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete

    filter of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set.

Discontinuous ultrapower characterization

A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda<j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. In other words, the embedding is discontinuous at all such $\lambda$.

Discontinuous embedding characterization

A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda<j(\lambda)$.

Ketonen characterization

An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^\ltkappa$. An ultrafilter $\mu$ on a cardinal $\lambda$ is uniform if all final segments $[\beta,\lambda)= \{\alpha<\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality.

Regular ultrafilter characterization

An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set. An ultrafilter $\mu$ is $(\kappa,\theta)$-regular if it is $\kappa$-complete and there is a family $\set{X_\alpha\st\alpha<\theta}\of \mu$ such that $\intersect_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.