Applied core model theory I

Inner model theory and core model theory might seem like their own niche in set theory, where you have to invest hundreds of hours just to get a glimpse of what’s going on. But behind all the complicated theory there are theorems in inner model theory which can be applied in many contexts with minimal background knowledge of the intricate technicalities appearing in their proofs. In this and the next couple of blog posts I’ll introduce one such theorem, explain how to use it, do a few mainstream set theory applications of it, and also provide a proof of it. Everything aside from the proof should hopefully be accessible to set theorists who aren’t inner model theorists.

core

The theorem I mentioned above is a core model dichotomy theorem, which, in the version I’ll present it, can be used to show that various hypotheses imply that projective determinacy holds; i.e. that every projective set of reals is determined. In terms of large cardinals this is equivalent to having certain proper class models M_n(x) for x\in\mathbb R, with x\in M_n(x) and having n Woodin cardinals. This is not just a consistency result: it shows that the hypotheses directly gives the determinacy and the models with the Woodin cardinals.

For this first post, let’s introduce some of the objects that we’re going to be using. As the theorem has the word core model in it, let’s start there. I’ve talked about the core model in a previous blog post, but that post was mostly directed towards inner model theorists, so I’m not really going to repeat anything from that post. Instead of talking about precisely what the core model is, I’d rather focus on the key properties that we often want to use in applications of core model theory: these are \textsf{GCH}covering and generic absoluteness, all of which are simple to state and yet incredibly powerful.

The core model is a proper class structure which is denoted by K, which I guess is for arbitrary reasons, like V and L. The covering property (or weak covering property, as it’s also called) states that whenever \kappa is a singular cardinal in V then \kappa^{+K}=\kappa^{+V}, and the generic absoluteness property states that K stays the same when we move to (set) forcing extensions.

The covering property is often presented as witnessing that “V is close to K“; if \textsf{GCH} holds then this can be seen by the fact that they both have the same amount of bijections f\colon\kappa\to\kappa, since in this case \kappa^{+V}=(\kappa^\kappa)^V (and \kappa^{+K}=(\kappa^\kappa)^K always holds since K\models\textsf{GCH}). The generic absoluteness property can be seen as K being canonical somehow, that its construction doesn’t depend on its surroundings.

We can also build the relativised core model K(x) for a set x, which satisfies that x\in K(x), covering holds above the rank of x, and generic absoluteness holds for all universes that have x as an element.  The theorem is then the following, where we recall that a forcing extension is \theta-small if the forcing poset has size {<}\theta.

The PD dichotomy. Assume that the core model K(x)|\theta doesn’t exist for some x\in H_\theta and some \theta>\omega either being a \beth-fixed point or \theta=\infty. Then \textsf{PD}, projective determinacy, holds.

Next time we’ll look at some concrete applications of this dichotomy, before we start digging into the proof. Stay tuned!

Here’s a link to the follow-up post