MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

In $L(\mathbb{R})$ we have $AD$ and it is a powerful tool to prove theorems. Almost all of the theorem proved with $AD$ come in a very natural way: we use games and determinacy as in the First/Second/Third Periodicity Theorems. However no "$ZFC$+Large $ZF$+Large Cardinal" proof is known for the Periodicity Theorems. Another example is that of the Perfect Set Property: Using $AD$ all sets of reals have the Perfect Set Property, but is a proof of the statement "Assuming infinitely many Woodin cardinals with a measurable above then every set of reals has the perfect set property" known?