Martin's maximum\(^{++}\) implies Woodin's axiom \((*)\)
In this paper, the authors provide a deep connection between two competing strong maximality prinicples: Martin's maximum and Woodin's \((*)\), showing that a strengthening of the former implies the latter. Martin's maximum is a forcing axiom which was introduced in [\textit{M. Foreman} et al., Ann. Math. (2) 127, No. 1, 1--47 (1988; Zbl 0645.03028)]. It states that for every forcing notion \(\mathbb{P}\) that preserves stationarity of subsets of \(\aleph_1\), and a collection of \(\aleph_1\) many dense open sets, \(\mathcal{D}\), there is a filter \(g \subseteq \mathbb{P}\), such that \(\forall D \in \mathcal{D}, g \cap D \neq \emptyset\). The axiom \(\mathrm{MM}^{++}\) is a strengthening of \(\mathrm{MM}\), in which we also require the filter to realize \(\aleph_1\) many names for stationary subsets of \(\omega_1\) as stationary sets. Woodin's axiom \((*)\) is the statement that the axiom of determinacy holds in \(L(\mathbb{R})\) and there is an \(L(\mathbb{R})\)-generic filter \(g\) for the forcing \(\mathbb{P}_{\max}\) such that \(P(\omega_1)\subseteq L(\mathbb{R})[g]\). The forcing \(\mathbb{P}_{\max}\) and the axiom \((*)\) first appeared in [\textit{W. H. Woodin}, The axiom of determinacy, forcing axioms, and the nonstationary ideal. Berlin: Walter de Gruyter (1999; Zbl 0954.03046)]. In this book, Woodin also derived many consequences of \((*)\). The forcing \(\mathbb{P}_{\max}\) is homogeneous, and thus every statement that holds in \(L(\mathbb{R})[g]\) is forced by the trivial condition and thus equivalent to a statement in \(L(\mathbb{R})\). In particular, the theory of \(H(\omega_2)\) under \((*)\) cannot be changed by set forcing, once large cardinals exist. This paper resolves the tension between those two principles by showing that \(MM^{++}\) implies \((*)\). Since \(MM^{+\omega}\) does not imply \((*)\), as was illustrated by \textit{P. B. Larson} [Ann. Pure Appl. Logic 156, No. 1, 110--122 (2008; Zbl 1153.03035) ] this result is optimal. Let me give a few details regarding the proof. Under some mild conditions, every set \(A \subseteq \omega_1\) corresponds to a filter, \(g_A \subseteq \mathbb{P}_{\max}\), that consists of all conditions \(\langle M, I, a\rangle\in\mathbb{P}_{\max}\) for which there is a generic iteration of length \(\omega_1\) to a model \(\langle M_{\omega_1}, NS_{\omega_1} \cap M_{\omega_1}, A\rangle\). In order to show that this filter is generic, we must find for every dense open set \(D\) a condition \(q \in D \cap g_A\). The main lemma in the proof shows that there is a forcing \(\mathbb{P}\) that preserves stationary subsets of \(\omega_1\), and adds a \(\mathbb{P}_{\max}\)-condition \(p \in D\), together with a generic iteration of it witnessing \(p \in g_A\). Using \(MM^{++}\), one can find a sufficiently generic filter for this forcing and construct the condition \(p\). The conditions of the forcing \(\mathbb{P}\) are finite pieces of information (given by sentences in an extended language) about the \(\mathbb{P}_{\max}\)-condition, the generic iteration of it, the description of \(D\) as well as submodels of \(H(\omega_3)\) that insures the preservation of stationary sets. This forcing can be viewed as an improved version of the forcing of \textit{B. Claverie} and \textit{R. Schindler} for making \(u_2 = \omega_2\) [J. Symb. Log. 74, No. 1, 187--200 (2009; Zbl 1163.03026)]. In order for a finite collection of sentences to be a condition, it must be \emph{certified} by an actual iteration in the generic extension by \(\mathrm{Col}(\omega,\omega_2)\). This assertion verifies that the generic object is well behaved. A universally Baire representation of the dense open set \(D\) is used in order to show that even though each finite collection of sentences corresponds to a different certificate, the generic object is going to be in \(D\). The paper is well written and the authors put a lot of effort into conveying the intuition behind the construction and the results of the paper.
scientific article; zbMATH DE number 7353242
scientific article