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If ℝ^# exists then why is cof(θ^L(ℝ)) = ω? Also I have the same question for the L(V_λ+1) generalization (if it's actually a different proof; I presume it isn't), i.e. if θ is defined as the sup of the surjections in L(V_λ+1) of V_λ+1 onto an ordinal, then if V_λ+1^# exists why is cof(θ^L(V_λ+1)) = ω?

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 $\textrm{cof}(\Theta^{L(V_{\lambda+1})}) = \omega$?

If R^# exists then why is cof(\theta^{L(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}^# exists why is cof(\theta^{L(V_{\lambda+1})}) = \omega?

If ℝ^# exists then why is cof(θ^L(ℝ)) = ω? Also I have the same question for the L(V_λ+1) generalization (if it's actually a different proof; I presume it isn't), i.e. if θ is defined as the sup of the surjections in L(V_λ+1) of V_λ+1 onto an ordinal, then if V_λ+1^# exists why is cof(θ^L(V_λ+1)) = ω?