# 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 $\theta\geq\kappa$. | ||

+ | |||

+ | 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. | + | 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 === | === 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 $|s|^M\lt j(\kappa)$. | 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 === | |

− | measure on $P_\kappa\theta$. | + | |

+ | 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. | 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 $|I|=\kappa$. |

## 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 $\kappa\leq\theta$.

## Contents

- 1 Strong compactness characterization
- 2 Strong compactness embedding characterization
- 3 Fine measure characterization
- 4 Cover property characterization
- 5 Filter extension characterization
- 6 Discontinuous ultrapower characterization
- 7 Discontinuous embedding characterization
- 8 Ketonen characterization
- 9 Regular ultrafilter 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. 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 $P_\kappa\theta$, isfineif 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 $|I|=\kappa$.