Let $V$ denote the von Neumann universe and $L$ Gödel's constructible universe. For any set $X$, let $P(X)$ denote the power set of $X$. Assume that $0^\sharp$ exists (and ZFC). What is the smallest ordinal $\alpha$ such that $L \cap P(L_{\alpha})$ is uncountable? (If $V = L$, then $\alpha = \omega$, but if $0^\sharp$ exists, then $\alpha > \omega$.) Login To add answer/comment