HODs of models of determinacy

HOD is the proper class of all sets x such that both x and all the elements of the transitive closure of x are definable using ordinal parameters. HOD is a model of ZFC, but in general its structure is not really known. In the late 90’s it was shown by Steel and Woodin that \textsf{HOD}^{L(\mathbb R)} exhibits mouse-like behaviour, and since then there’s been a great interest in finding the HODs of other models than L(\mathbb R). I’ll here give a (non-exhaustive) overview of both which HODs have been shown to have this mouse-like structure and also explain the general strategy used so far in finding these mice.

Let’s start off with some history. Interest in studying the structure of HOD started in the beginning of the 80’s, where three out of the fourteen famous Delfino problems involved HOD. Assuming AD+V=L(\mathbb R),

  • is every regular cardinal <\Theta measurable? (yes, shown by Moschovakis and Kechris)
  • is there a \kappa which is 2^\kappa-supercompact in HOD? (no, shown by Woodin and Shelah)
  • does HOD satisfy GCH? (yes, shown by Steel and Woodin)

The positive solution to the last question sparked a lot of interest, since Steel showed even more than just GCH – he showed the remarkable fact that, assuming AD^{L(\mathbb R)}, \textsf{HOD}^{L(\mathbb R)} is a mouse below \Theta, the least ordinal \alpha to which there is no surjection f:\mathbb R\to\alpha. This was the kickoff to a great deal of HOD research, and Woodin showed that the full HOD of L(\mathbb R) is a new kind of mouse, having both an extender sequence and a fragment of its own iteration strategy as predicates – this breed of mice is now called hod mice.

A natural question is then if this is something peculiar to L(\mathbb R) or if it also holds in other HODs. Investigating the HOD of L[x] turned out to be incredibly hard, but the model L[x,g] turned out to be more amenable to an approach analogous to Steel’s approach to L(\mathbb R), where g is L[x]-generic for collapsing the least inaccessible of L[x] to be \aleph_1. This has very recently been generalised by Sargsyan and Uhlenbrock to the HOD of M_n(x,g).

Abstracting away, we can also view L(\mathbb R) as simply a special case of a model satisfying \textsf{ZF}+\textsf{AD}^+, where \textsf{AD}^+ is an ostensibly stronger version of \textsf{AD}. Taking this approach, we can then beef up this theory and ask about the HOD of the least model satisfying that theory. This has resulted in a HOD analysis of the least model of the various \Theta-theories floating around.

That was a bit of an overview. Let’s now dig down into how these HODs are being investigated. As a representative example, let’s take a look at the simplest instance: the HOD of L[x,g]. Let’s assume (boldface) {\bf\Delta}^1_2-determinacy, which is equivalent to the existence of M_1^\sharp and x^\sharp for all reals x\in\mathbb R. Strictly speaking we could do with the weaker assumption that

L[x]\models"\Delta^1_2\text{-determinacy and there exists an inaccessible cardinal}",

but having M_1^\sharp makes the argument a bit more clean. Fix some x\in\mathbb R such that M_1^\sharp\in L[x]. Firstly, our hypothesis implies that M_1^\sharp has a unique iteration strategy \Sigma_0, so we can construct the direct limit of all \Sigma_0-iterates N of M_1^\sharp such that N|(\delta^N)^{+N}\in\text{HC}^{L(\mathbb R)}, where \delta^N is the Woodin cardinal of N. Call this direct limit M_\infty^+ and its Woodin cardinal \delta_\infty.

The problem with working with M_\infty^+ is that it doesn’t exist within L[x,g], so we want to build an internal directed system of mice. I’ll leave out the technical details, but roughly speaking the system consists of all countable mice inside of L[x,g] which looks like M_1^\sharp (remember we’re in L[x,g], so countable here means of size less than the first inaccessible of L[x]). This then yields the direct limit M_\infty\subseteq L[x,g].

To show that M_\infty is a well-founded model, we define an elementary map \sigma:M_\infty\to M_\infty^+, which suffices as M_\infty^+ is well-founded as it’s a \Sigma_0-iterate of M_1^\sharp. This map \sigma is defined as taking x\in M_\infty, pulling it back to a mouse which is in both directed systems (the existence of such a mouse has to be shown here) and then going up to M_\infty^+ via the direct limit map. We also get that \sigma\upharpoonright\delta_\infty+1=\text{id}, so that in particular the two direct limits have the same Woodin cardinal.

To recap, we’ve now got an internal limit M_\infty\subseteq L[x,g] and an external limit M_\infty^+ which agree below their common Woodin cardinal \delta_\infty. Since our internal directed system was definable in L[x,g], we get that

M_\infty\subseteq\textsf{HOD}^{L[x,g]}.

The next step is to show the opposite inclusion, for which we need a so-called derived model resemblance. First, let define \alpha^* for any ordinal \alpha to be the image of \alpha inside M_\infty, yielding a function F(\alpha):=\alpha^*. The derived model of M_\infty is M_\infty[h], where h is M_\infty-generic for collapsing the least inaccessible \kappa_\infty of M_\infty strictly above \delta_\infty to be \aleph_1. The derived model resemblance then implies that

L[x,g]\models\varphi[\alpha_1,\dots,\alpha_n]\Leftrightarrow M_\infty[h]\models\varphi[\alpha_1^*,\dots,\alpha_n^*]

for any formula \varphi and ordinals \alpha_1,\dots,\alpha_n. The next step is then to show that

\textsf{HOD}^{L[x,g]}=L[M_\infty,F]. (1)

We get the right-to-left inclusion simply because both M_\infty and F are L[x,g]-definable. For the other direction the first thing to show (which I’ll skip here) is that \textsf{HOD}^{L[x,g]}=L[A] for an L[x,g]-definable A\subset\aleph_2^{L[x,g]}. Given this fact we can then use the above resemblance to transfer the fact over to M_\infty, so that A lies in L[M_\infty,F], yielding (1).

Now let \Lambda be the restriction of \Sigma_0 to trees \mathcal T on M_\infty such that \mathcal T\in M_\infty|\kappa_\infty. The last step is then showing that

\textsf{HOD}^{L[x,g]}=L[M_\infty,\Lambda],

which is done by showing that \Lambda is definable using M_\infty and F, and then using (1). Summarising, HOD was found by

  1. Cooking up a directed system of mice
  2. Showing that we can transfer truths between L[x,g] and the derived model of the direct limit
  3. Establishing that HOD is of the form L[A] for some L[x,g]-definable A
  4. Using (2) and (3) to show that HOD is of the form L[M_\infty,F] for some F
  5. Defining a partial strategy of M_\infty from F.

This strategy is basically what’s going on in the L(\mathbb R) cases as well, and I suspect this is also what’s happening with minimal models of \Theta-theories.