## Square principles in Pmax extensions

As mentioned previously, I am part of a SQuaRE (Structured Quartet Research Ensemble), “Descriptive aspects of inner model theory”. The other members of the group are Paul Larson, Grigor Sargsyan, Ralf Schindler, John Steel, and Martin Zeman. We have just submitted our paper Square principles in ${\mathbb P}_{\rm max}$ extensions to the Israel Journal of Mathematics. The paper can be downloaded form my papers page, or from the arXiv.

The forcing ${\mathbb P}_{\rm max}$ was developed by Hugh Woodin in his book The axiom of determinacy, forcing axioms, and the nonstationary ideal${\mathbb P}_{\rm max}$ belongs to $L({\mathbb R})$ and, if determinacy holds, the theory that it forces is combinatorially rich, and we do not currently know how to replicate it with traditional forcing methods.

In particular, Woodin showed that, starting with a model of ${\sf AD}_{\mathbb R}+$$\Theta$ is regular”, a strong form of determinacy, the ${\mathbb P}_{\rm max}$ extension satisfies ${\sf MM}({\mathfrak c})$, the restriction of Martin’s maximum to posets of size at most ${}|{\mathbb R}|$. It is natural to wonder to what extent this can be extended. In this paper, we study the effect of ${\mathbb P}_{\rm max}$ on square principles, centering on those that would be decided by ${\sf MM}({\mathfrak c}^+)$.

These square principles are combinatorial statements stating that a specific version of compactness fails in the universe, namely, there is a certain tree without branches. They were introduced by Ronald Jensen in his paper on The fine structure of the constructible universe. The most well known is the principle $\square_\kappa$:

Definition. Given a cardinal $\kappa$, the principle $\square_{\kappa}$ holds iff there exists a sequence $\langle C_{\alpha} \mid \alpha < \kappa^{+} \rangle$ such that for each $\alpha < \kappa^{+}$,

1. Each $C_{\alpha}$ is club in $\alpha$;
2. For each limit point $\beta$ of $C_{\alpha}$, $C_{\beta} = C_{\alpha} \cap \beta$; and
3. The order type of each $C_\alpha$ is at most $\kappa$.

For $\kappa=\omega$ this is true, but uninteresting. The principle holds in Gödel’s $L$, for all uncountable $\kappa$. It is consistent, relative to a supercompact cardinal, that it fails for all uncountable $\kappa$. For example, this is a consequence of Martin’s maximum.

Recently, the principle $\square(\kappa)$ has been receiving some attention.

Definition. Given an ordinal $\gamma$, the principle $\square(\gamma)$ holds iff there exists a sequence $\langle C_{\alpha} \mid \alpha <\gamma \rangle$ such that

1. For each $\alpha < \gamma$, each $C_{\alpha}$ is club in $\alpha$;
2. For each $\alpha<\gamma$, and each limit point $\beta$ of $C_{\alpha}$, $C_{\beta} = C_{\alpha} \cap \beta$; and
3. There is no thread through the sequence, i.e., there is no club $E\subseteq \gamma$ such that $C_{\alpha} = E \cap \alpha$ for each limit point $\alpha$ of $E$.

Using the Core Model Induction technique developed by Woodin, work of Ernest Schimmerling, extended by Steel, has shown that the statement

$2^{\aleph_1}=\aleph_2+\lnot\square(\omega_2)+\lnot\square_{\omega_2}$

implies that determinacy holds in $L({\mathbb R})$, and the known upper bounds in consistency strength are much higher.

Here are some of our results: First, if one wants to obtain $\lnot\square_{\omega_2}$ in a ${\mathbb P}_{\rm max}$ extension, one needs to start from a reasonably strong determinacy assumption:

Theorem. Assume ${\sf AD}_{\mathbb R}+$$\Theta$ is regular”, and that there is no $\Gamma \subseteq \mathcal{P}(\mathbb{R})$ such that $L(\Gamma, \mathbb{R}) \models$$\Theta$ is Mahlo in ${\sf HOD}$“. Then $\square_{\omega_{2}}$ holds in the ${\mathbb P}_{\rm max}$ extension.

This results uses a blend of fine structure theory with the techniques developed by Sargsyan on his work on hybrid mice. The assumption cannot be improved since we also have the following, the hypothesis of which are a consequence of  ${\sf AD}_{\mathbb R}+V=L({\mathcal P}({\mathbb R}))+$$\Theta$ is Mahlo in ${\sf HOD}$”.

Theorem. Assume that ${\sf AD}^+$ holds and that $\theta$ is a limit on the Solovay sequence such that that there are cofinally many $\kappa<\theta$ that are limits of the Solovay sequence and are regular in ${\sf HOD}$. Then $\square_{\omega_2}$ fails in the ${\mathbb P}_{\rm max}$ extension of ${\sf HOD}_{\mathcal{P}_{\theta}(\mathbb{R})}$.

Here, ${\mathcal P}_\alpha({\mathbb R})$ denotes the collection of subsets of ${\mathbb R}$ of Wadge rank less than $\alpha$. The Solovay sequence, introduced by Robert Solovay in The independence of ${\sf DC}$ from ${\sf AD}$, is a refinement of the definition of $\Theta$, the least ordinal $\alpha$ for which there is no surjection $f:{\mathbb R}\to\alpha$:

Definition. The Solovay sequence if the sequence of ordinals $\langle \theta_{\alpha} \mid \alpha \leq \gamma \rangle$ such that

1. $\theta_{0}$ is the least ordinal that is not the surjective image of the reals by an ordinal definable function;
2. For each $\alpha < \gamma$, $\theta_{\alpha + 1}$ is the least ordinal that is not the surjective image of the reals by a function definable from an ordinal and a set of reals of Wadge rank $\theta_{\alpha}$;
3. For each limit ordinal $\beta \leq \gamma$, $\theta_{\beta} = \sup\{\theta_{\alpha} \mid \alpha < \beta\}$; and
4. $\theta_{\gamma} = \Theta$.

The problem with the result just stated is that choice fails in the resulting model. To remedy this, we need to start with stronger assumptions. Still, these assumptions greatly improve the previous upper bounds for the consistency (with ${\sf ZFC}$) of $2^{\aleph_1}=\aleph_2+\lnot\square(\omega_2)+\lnot\square_{\omega_2}$. In particular, we now know that this theory is strictly weaker than a Woodin limit of Woodin cardinals.

Theorem. Assume that ${\sf AD}_{\mathbb R}$ holds, that $V = L(\mathcal{P}(\mathbb{R}))$, and that stationarily many elements $\theta$ of cofinality $\omega_{1}$ in the Solovay sequence are regular in ${\sf HOD}$. Then in the ${\mathbb P}_{\rm max} * {\rm Add}(\omega_{3},1)$-extension, $\square_{\omega_2}$ fails.

(The forcing ${\rm Add}(\omega_3,1)$ adds a Cohen subset of $\omega_3$. This suffices to well-order ${\mathcal P}({\mathbb R})$, and therefore to force choice. That in the resulting model we also have $2^{\aleph_1}=\aleph_2+\lnot\square(\omega_2)$ follows from prior work of Woodin.)

Finally, I think I should mention a bit of notation. In the paper, we say that $\kappa^+$ is square inaccessible iff $\square_\kappa$ fails. We also say that $\gamma$ is threadable iff $\square(\gamma)$ fails. This serves to put the emphasis on the negations of the square principles, which we feel is where the interest resides. It also solves the slight notational inconvenience of calling $\square_\kappa$ a principle that is actually a statement about $\kappa^+$.