Covering property

The covering property is a property of inner models, in particular core models, which is a measure of how "closely they approximate" the real universe \(V\) of sets. Namely, we say an inner model \(N\) has the covering property iff it is able to "cover" uncountable sets of ordinals: for every uncountable set \(X\) of ordinals, there is \(Y \in N\) so that \(X \subset Y\) and \(|X| = |Y|\). Of course, \(V\) has the covering property, and so \(V = L\) implies all inner models have the covering property.

It is known that the existence of \(0^\sharp\) is equivalent to \(L\) not having the covering property and, in general, the existence of the sharp for \(N\) is equivalent to \(N\) not having the covering property.

The covering property is intrinsically related to cardinal-related "closeness" to the universe. Namely, \(N\) has the covering property iff there is some singular cardinal \(\lambda\) which is singular in \(L\) or there is some successor cardinal \(\lambda\) which is a limit cardinal in \(L\); iff there is some singular cardinal \(\lambda\) so that, \((\lambda^+)^L < \lambda^+\).

This hierarchy can be refined and stratified: we say \(N\) has the \(\delta\)-covering property, for a cardinal \(\delta\), if, for every \(X \subseteq N\) with \(|X| < \delta\), there is \(Y \in N\) so that \(X \subset Y\) and \(|Y| < \delta\). Every inner model has the \(\omega\)-covering property, and having the covering property is equivalent to having the \(\delta\)-covering property for all \(\delta > \omega_1\). This stratification is important and related to weak extender models for supercompactness.

Namely, if \(\delta\) is a supercompact cardinal and \(N\) is a weak extender model for \(\delta\)'s supercompactness, then:

• \(N\) has the \(\delta\)-covering property
• If \(\lambda > \delta\) is regular in \(N\), then the cardinality of \(\lambda\) is regular.
• If \(\lambda > \delta\) is singular, then \(\lambda\) is singular in \(N\) and \((\lambda^+)^N = \lambda^+\).

This shows a paradigm shift, in that weak extender models for supercompactness must be close to the universe, regardless of what further sharps and large cardinals exist.