Martin's maximum\(^{++}\) implies Woodin's axiom \((*)\)
type: Item
instance of: scholarly article
title: Martin's maximum\(^{++}\) implies Woodin's axiom \((*)\)
zbMATH Open document ID: 1496.03203
author: David Asperó
author: Ralf Schindler
published in: Annals of Mathematics. Second Series
publication date: 2021-06-01T00:00:00Z
review text: 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…
reviewed by: Yair Hayut
Mathematics Subject Classification ID: 03E57
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E60
zbMATH DE Number: 7353242
zbMATH Keywords: continuum hypothesis
zbMATH Keywords: forcing axioms
zbMATH Keywords: Martin's maximum
zbMATH Keywords: \(\mathbb{P}_{\max}\) forcing
zbMATH Keywords: axiom \((*)\)
Wikidata QID: Q113999272
MaRDI profile type: Publication
full work available at URL: annals.2021.193.3.3
OpenAlex ID: W3159191542
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DOI: 10.4007/ANNALS.2021.193.3.3
description: scientific article; zbMATH DE number 7353242