If ℝ# exists then why is cof(θL(ℝ)) = ω? Also I have the same question for the L(Vλ+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(Vλ+1) of Vλ+1 onto an ordinal, then if Vλ+1# exists why is cof(θL(Vλ+1)) = ω? 



Cofinality of Theta if sharps exist
This is because the pieces of the sharp singularize Theta. Let s\_n be the sequence of the first n cardinals above continuum and let a\_n be the nth cardinal above continuum. Then the theory of reals with a parameter s\_n in L\_{a\_n+1}(R) is a set of reals A\_n. They are Wadge cofinal in Theta, another words the sequence is not in L(R) but each A\_n is and that is why you get a singularization.


