TITLE: Cofinality of Theta if sharps exist
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: Scott, the best way to think of sharps is via mice. Think of x^# as a mouse over x with one measure which is iterable. R^# is a mouse over R with one measure which is iterable. Things become very easy ones you make the move from sharps as reals or sets of reals or etc to sharps as mice.