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

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

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