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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^\
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