Zero sharp

Description

Let $L$ be the constructible universe. Suppose that there is a closed unbounded class $I$ of ordinals with the following property for every uncountable cardinal $\kappa$. The set $I \cap \kappa$ has order type $\kappa$ and is a set of indiscernibles for the structure $(L_\kappa, \in)$. Every element of $L_\kappa$ is definable in $L_\kappa$ from finitely many elements of $I \cap \kappa$. Such a class is unique when it exists, and its elements are the Silver indiscernibles. In this case zero sharp is the set $$0^\sharp = \set{\ulcorner \varphi \urcorner : L_{\aleph_\omega} \models \varphi[\aleph_1, \ldots, \aleph_n]}.$$ Here $\varphi(x_1, \ldots, x_n)$ runs over the formulas of set theory, and $\ulcorner \varphi \urcorner$ is the Gödel number of $\varphi$. One says that $0^\sharp$ exists when such a class $I$ exists.