If $\mathbb R^\sharp$ exists then why is $\textrm{cof}(\Theta^{L(\mathbb R)})=\omega$? Also I have the same question for the $L(V_{\lambda+1})$ generalization (if it's actually a different proof; I presume it isn't), i.e. if $\Theta$ is defined as the sup of the surjections in $L(V_{\lambda+1})$ of $V_{\lambda+1}$ onto an ordinal, then if $V_{\lambda+1}^\sharp$ exists why is $\textrm{cof}(\Theta^{L(V_{\lambda+1})}) = \omega$?