Sharp: Difference between revisions

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 \(0^\Ptext{¶}\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|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, \(0^\Ptext{¶}\) is an inner model for a [[Strong cardinal|strong]] cardinal.