Sharp

A sharp for an inner model \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of ordinals may not be able to be covered by sets in \(N\). For example, the sharp for \(L\) is \(0^\sharp\). In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't.

Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have.

One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^{sword}\) and \(0^\text{¶}\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's measurability. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^{sword}\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial Mitchell rank, i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\text{¶}\) is an inner model for a strong cardinal.