# Difference between revisions of "Strongly compact"

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=== Fine measure characterization === | === Fine measure characterization === | ||

− | An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete fine | + | An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete fine [[filter]] 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 === | === Cover property characterization === | ||

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=== Filter extension characterization === | === 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. | + | 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 === | === Discontinuous ultrapower characterization === | ||

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== Strongly compact cardinals and forcing == | == Strongly compact cardinals and forcing == | ||

− | If there is proper class-many strongly compact cardinals, each | + | If there is proper class-many strongly compact cardinals, then there is a model of $ZF$ + "all uncountable cardinals are singular". If each strongly compact cardinal is a limit of measurable cardinals, and if the limit of any sequence of strongly compact cardinals is singular, then there is a forcing extension V[G] that is a symmetric model of ZF + "all uncountable cardinals are singular" + "every uncountable cardinal is both almost [[Ramsey]] and a [[Rowbottom]] cardinal carrying a Rowbottom filter". |

− | cardinal carrying a Rowbottom filter". | + | |

This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals. | This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals. | ||

## Revision as of 01:20, 10 October 2017

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 elementary 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 $M^\kappa\subset M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.

## Contents

- 1 Diverse characterizations
- 1.1 Strong compactness characterization
- 1.2 Strong compactness embedding characterization
- 1.3 Fine measure characterization
- 1.4 Cover property characterization
- 1.5 Filter extension characterization
- 1.6 Discontinuous ultrapower characterization
- 1.7 Discontinuous embedding characterization
- 1.8 Ketonen characterization
- 1.9 Regular ultrafilter characterization

- 2 Strongly compact cardinals and forcing
- 3 Relation to other large cardinal notions

## Diverse characterizations

There are diverse equivalent characterizations of the strongly compact cardinals.

### 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 $\vec x=\langle x_\alpha\mid\alpha<\delta\rangle$ 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 filter 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 $t\subset M$ of size $\theta$ there is $s\in M$ with $t\subset s$ 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 $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. 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 $\{X_\alpha\mid\alpha<\theta\}\subset \mu$ such that $\bigcap_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.

## Strongly compact cardinals and forcing

If there is proper class-many strongly compact cardinals, then there is a model of $ZF$ + "all uncountable cardinals are singular". If each strongly compact cardinal is a limit of measurable cardinals, and if the limit of any sequence of strongly compact cardinals is singular, then there is a forcing extension V[G] that is a symmetric model of ZF + "all uncountable cardinals are singular" + "every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter". This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals.

## Relation to other large cardinal notions

Strongly compact cardinals are measurable. The least strongly compact cardinal can be equal to the least measurable cardinal, or to the least supercompact cardinal, by results of Magidor. (It cannot be equal to both at once because the least measurable cardinal cannot be supercompact.)

It is not currently known whether the existence of a strongly compact cardinal is equiconsistent with the existence of a supercompact cardinal.