Supercompact

Supercompact cardinals are a kind of large cardinal with powerful reflection properties. The construction of an inner model accommodating a supercompact cardinal is considered the holy grail of inner model theory, and is extremely difficult. Formally, a cardinal \(\kappa\) is called \(\lambda\)-supercompact iff there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). In particular, any measurable cardinal is \(\kappa\)-supercompact, and any \(\gamma\)-supercompact cardinal is \(\gamma\)-strong. Like how measurable cardinals can alternatively be defined in terms of ultrafilters, so can supercompact cardinals: \(\kappa\) is \(\lambda\)-supercompact iff there is a \(\kappa\)-complete, normal, fine ultrafilter on \([\lambda]^{< \kappa}\), the set of subsets of \(\lambda\) with size less than \(\kappa\).

Then \(\kappa\) is supercompact iff it is \(\lambda\)-supercompact for all \(\lambda > \kappa\). As previously mentioned, any supercompact cardinal is strong (and therefore has Mitchell rank \((2^\kappa)^+\)), as well as having many such cardinals below it. Any supercompact cardinal is also Woodin and a limit of Woodin cardinals, and much more.

Supercompact cardinals possess curious reflection properties that can explain their size - if a cardinal with some property that is witnessed by a structure of limited rank exists above a supercompact cardinal, then a cardinal with that property exists below that same cardinal. For example, the least \(n\)-huge cardinal, if it exists, is always less than the least supercompact cardinal; and if the generalized continuum hypothesis holds below the least supercompact cardinal, it holds everywhere. This is because the failure of GCH at a cardinal \(\nu\) can be witnessed within \(V_{\nu+2}\), and thus the existence of such a \(\nu\) above a supercompact implies its existence below a supercompact.

The least supercompact cardinal in particular possesses a potent \(\Pi^1_1\)-reflection property: the least supercompact is precisely the least \(\kappa\) so that, for every structure \(\mathcal{M}\) whose domain has cardinality at least \(\kappa\) and for every \(\Pi^1_1\)-sentence it satisfies, there is some \(\mathcal{N}\) which also satisfies that sentence so that the cardinality of the domain of \(\mathcal{N}\) is less than the cardinality of the domain of \(\mathcal{N}\), and whose relations are the restrictions of the relations of \(\mathcal{M}\) to the domain of \(N\). For example, if \(\varphi\) is a formula, \(R \subseteq V_\kappa\) and \((V_\kappa, \in, R) \models \varphi\) then there is some \(M\) so that \(|M| < \kappa\) and \((M, \in, R \cap M) \models \varphi\) - this already implies inaccessibility.