I’ve previously covered Woodin’s genericity iterations, being a method to “catch” any real using Woodin cardinals. Roughly, given any countable mouse M and a real x, we can iterate M to a model over which x is generic. An application of this is the phenomenon that Woodins present in mice allows them to be more projectively aware.
To explain what I mean by this, we need a definition, this one taken from the core model induction book.
Definition. Let , M a countable mouse, an uncountable cardinal of M and . Then understands A at if whenever P is an iterate of M and is -generic over P then . We also say that M understands A at if there exists such a , and simply that M understands A if there exist such and .
If a given set A is understood by M at some then M suddenly has an oracle which can inform it about membership of A, even though A might not be an element of M. Namely, it can ask of a real whether there exists a such that , where witnesses that M understands A.
The result concerning Woodin cardinals is then the following, essentially saying that the more Woodin cardinals M thinks there are, the more projective sets it knows of as well.
Theorem. Let M be a countable mouse and let be a Woodin cardinal of M. Then whenever M understands at , M also understands at any which is uncountable in M.
Proof. Let witness that M understands B at and define as follows. Let iff , is a name for a real and
To show that witnesses that M understands at we may firstly assume in the definition of understanding, to ease notation. Let be -generic over M. We have to show that
We start with the direction. Let and pick such that . Now, since is Woodin in M we can form a genericity iteration such that , where is -generic over P.
Furthermore, we can pick to be as large as we want in a genericity iteration, so by choosing it sufficiently large we may assume that and are fixed by . Because then , where is the iteration derived from . We thus have the following picture:
As witnesses that M understands B we have that , so that . This means that , implying that and then elementarity of yields that
concluding that by construction of . As for the direction, if then holds, so that , i.e. that and we’re done. QED
Note that being understood is closed under negation, so that if is understood then so is . Also note that since sets of reals are absolute between mice, every set is understood by any M and . By assuming we got a Woodin lying around we can then catch real parameters, making sure that this is still true for boldface sets of reals and hence also for boldface sets by the above Theorem.
In particular, this means that for every set A. The theorem then implies that whenever M has Woodins then this holds true for sets, and that if it has a limit of Woodins then it’s true for all projective sets. This is not quite as strong as being projectively correct, as we might have projective sets which have empty intersection with M. To ensure correctness we need something stronger, called Suslin capturing, which we’ll cover at some later point.
But of course we don’t have to restrict ourselves to the projective hierarchy. Whenever M understands any class , Woodins present inside M will ensure that M can reason about the corresponding “ projective hierarchy”.