Martin's Maximum and the \(\mathbb{P}_{\max}\) Axiom \((*)\)
Trying to settle the compatibility problem of the saturation of the nonstationary ideal on \(\omega_1\) and the Continuum Hypothesis, \textit{W. H. Woodin} initiated an analysis of the inner model \(L(P(\omega_1))\) under suitable large cardinal hypotheses in his 934 pages monograph [The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications. 1, de Gruyter, Berlin (1999; Zbl 0954.03046)]. The core of the analysis is a partial order, denoted by \({\mathbb P}_{\max }\), in \(L(R)\), which is nontrivial under the axiom of determinacy, \(\text{AD}^{L(R)}\). Analysis of \(L(P(\omega_1))\) essentially reduces to an axiom, called Axiom \((*)\), which states that the Axiom of Determinacy holds in the inner model \(L(R)\) and the inner model \(L(P(\omega_1))\) is a generic extension of \(L(R)\) by the forcing notion \({\mathbb P}_{\max}\). Woodin's analysis indicates that there are various connections and similarities between the Axiom \((*)\) and a well-known forcing axiom, Martin's Maximum, due to Foreman, Magidor and Shelah (Martin's maximum, saturated ideals and nonregular ultrafilters, Ann. of Math. 127, 1-47(1988)), which states that if \(P\) is a stationary set preserving partial order and \(\{D_\alpha \mid \alpha<\omega_1\}\) is a sequence of dense subsets of \(P\), then there exists a filter meeting all the \(D_{\alpha }\)'s. For example, both Axiom \((*)\) and Martin's Maximum imply that the Axiom of Choice holds in \(L(P(\omega_1))\), the nonstationary ideal on \(\omega_1\) is saturated, \(2^{\aleph_0} = \aleph_2\), etc. Naturally, questions relating these two axioms occur. One of them is settled in the paper under review. Namely, it is shown that relative to the consistency of the existence of supercompact limit of supercompact cardinals, it is consistent that Martin's Maximum holds but Axiom \((*)\) fails. The proof applies a generalization of Woodin's argument, which is in Woodin's book used to show that a strong bounded form of Martin's Maximum does not imply Axiom \((*)\), and a rework of the original consistency proof of Martin's Maximum, due to Foreman, Magidor and Shelah. Still, major questions relating the two axioms remain open.
scientific article; zbMATH DE number 1546601