>[!info] Definition > A class of ordinals $I$ are *indiscernibles* for the model $\mathfrak{A}$ if for every $n \in \omega$, and every formula $\varphi\left(v_1, \ldots, v_n\right)$, $\mathfrak{A} \vDash \varphi\left[\alpha_1, \ldots, \alpha_n\right] \quad$ if and only if $\mathfrak{A} \vDash \varphi\left[\beta_1, \ldots, \beta_n\right]$ whenever $\alpha_1<\ldots<\alpha_n$ and $\beta_1<\ldots<\beta_n$ are two increasing sequences of elements of $I$. ## Silver indiscernibles A class of ordinals $I$ is a class of *Silver indiscernibles* if it is a closed unbounded class containing all uncountable cardinals such that for every uncountable cardinal $\kappa$ : (a) $|I \cap \kappa|=\kappa$, (b) $I \cap \kappa$ is a set of indiscernibles for $\left(L_\kappa, \in\right)$, and (c) every $a \in L_\kappa$ is definable in $\left(L_\kappa, \in\right)$ from $I \cap \kappa$. **Theorem (Silver).** If there is a [[Ramsey]] cardinal, then 1. For every uncountable cardinals $\kappa<\lambda$, $(L_{\kappa},\in)\prec (L_{\lambda},\in)$; 2. There is a unique class $I$ as above. The statement "$0^{\sharp}$ [[Zero sharp|exists]]" is defined as the conjunction 1+2.