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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 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 $\kappa\leq\theta$.
There are diverse equivalent characterizations of the strongly compact cardinals.
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.
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 $|s|^M\lt j(\kappa)$. (Kanamori, 2009)
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 $|s|^M<j(\kappa)$.
An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a fine measure on $\mathcal{P}_\kappa(\theta)$. The notation $\mathcal{P}_\kappa(\theta)$ means $\{\sigma\subset\theta\mid |\sigma|<\kappa\}$. (Kanamori, 2009)
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. (Kanamori, 2009)
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 $\lambda$.
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 $\sup j’’\lambda<j(\lambda)$.
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.
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 $|I|=\kappa$.
If there is proper class-many strongly compact cardinals, then there is a generic model of $\text{ZF}$ + “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 $\text{ZF}$ + “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.
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. (Jech, 2003) (It cannot be equal to both at once because the least measurable cardinal cannot be supercompact.)
Even though strongly compact cardinals imply the consistency of the negation of the singular cardinal hypothesis, SCH, for any singular strong limit cardinal $\kappa$ above the least strongly compact cardinal, $2^\kappa=\kappa^+$ (also known as “SCH holds above strong compactness”). (Jech, 2003)
If there is a strongly compact cardinal $\kappa$ then for all $\lambda\geq\kappa$ and $A\subseteq\lambda$, $\lambda^+$ is ineffable in $L[A]$.
It is not currently known whether the existence of a strongly compact cardinal is equiconsistent with the existence of a supercompact cardinal. The ultrapower axiom gives a positive answer to this, but itself isn’t known to be consistent with the existence of a supercompact in the first place.
Every strongly compact cardinal is strongly tall, although the existence of a strongly compact cardinal is equiconsistent with “the least measurable cardinal is the least strongly compact cardinal, and therefore the least strongly tall cardinal”, so it could be the case that the least of the measurable, tall, strongly tall, and strongly compact cardinals all line up.
Strongly compact cardinals are related to the topological notion of compactness, interestingly enough.
A topological space $X$ is called $\kappa$-compact when every open cover has a subcover of size below $\kappa$. More intuitively, it “looks” as though it has size below $\kappa$. For example, the $\aleph_0$-compact subspaces of the real number line are just the subspaces which are bounded. For example, a shape with finite area could be considered $\aleph_0$-compact, even though the amount of points is not only infinite but continuum-sized.
The product of a collection of spaces is a little difficult to describe intuitively. However, it notably increases the amount of “dimensions” to a space. For example, the product of $n$-copies of the real number line is just the $n$-dimensional euclidean space (the line, the plane, etc.). Also, the general intuition is that it doesn’t make spaces any bigger than the biggest one in the collection, so the product of a bunch of small spaces and a big space should be no ‘bigger’ than the big space.
The idea is that the product of $\kappa$-compact spaces should itself be $\kappa$-compact, since the product doesn’t make spaces any “bigger.” However, there are examples of two $\aleph_1$-compact spaces (they “look countably infinite”) which combine to make a space which isn’t $\aleph_1$-compact (“looks uncountable”). However, if $\kappa$ is strongly compact, then this intuition holds; the product of any $\kappa$-compact spaces is strongly compact. One could maybe see why strongly compact cardinals are so big then; they imply that combining a bunch of small-relative-to-$\kappa$ spaces together by adding arbitrarily many dimensions keeps the space looking small relative to $\kappa$.’
Tychonoff’s theorem is precisely the statement that the product of $\aleph_0$-compact spaces is $\aleph_0$-compact; that is, if you combine a bunch of finite-looking spaces together and keep adding more and more dimensions, you get a space which is finite-looking.
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