# Strongly compact cardinal

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 $\theta\geq\kappa$.

### 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.

### 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 $P_\kappa\theta$.

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 $|s|^M<j(\kappa)$.
Strong compactness 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 $\kappa\leq\lambda\leq\theta^{\lt\kappa}$.
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 $\sup j''\lambda<j(\lambda)$.
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 $\kappa\leq\lambda\leq\theta^\ltkappa$.
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.

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.

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 $\kappa$ 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 $\kappa\leq\theta$.

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 $|I|=\kappa$.