I’ve previously mentioned an interesting characterisation of Woodin cardinals, that they are more or less characterised by the fact that we can do genericity iterations with them, i.e., that we can “capture reals” when Woodin cardinals are present. This is an exercise in the core model induction book, and Stefan Mesken recently found a way to solve this (see his solution here), which I’ll be presenting here.
I’ve previously introduced the notion of absorbing reals, but let’s recall it here for convenience:
Definition. Let M be a premouse, an iteration strategy for M and . Then we say that absorbs reals at if there exists an iteration by living on such that for some P-generic . We also say that absorbs reals at if it absorbs reals at for all .
The characterisation is then this, which is Exercise 1.4.5 in the core model induction book:
Proposition. Let M be a premouse, such that exists. Then absorbs reals at iff is either a Woodin cardinal in M or a limit of Woodin cardinals in M.
Proof. The backwards direction is by using genericity iterations, which I’ve previously covered, so we show the forwards direction. Firstly we may assume that M is countable, as otherwise we could take a countable hull and work with N and its pulled back version of .
Assume that is not Woodin in M; we’ll show that it’s a limit of Woodins in M. Let be an ordinal and define to be the least ordinal which absorbs reals at . We will show that is in fact Woodin in M, so assume it’s not. Since Woodins always absorb reals we get that there are no Woodins in the interval in M.
As M doesn’t have any Woodins in the interval we get that any -iteration living on is guided by Q-structures, so that can define as
For more information about Q-structures, see e.g. John Steel’s handbook chapter or my MSc thesis. The reason why the Q-structures exist in the generic extension is by using absoluteness of wellfoundedness.
Now let be an iteration tree on M living on with last model P such that with P-generic. Then by the above. But is cofinal in , making it singular in P[g]. But it’s regular in P and has the -cc, so it’s also regular in P[g], a contradiction! So is Woodin in M, and is thus a limit of Woodins in M. QED