PFA and ideals on \(\omega_{2}\) whose associated forcings are proper
type: Item
instance of: scholarly article
title: PFA and ideals on \(\omega_{2}\) whose associated forcings are proper
zbMATH Open document ID: 1253.03078
DOI: 10.1215/00294527-1716793
published in: Notre Dame Journal of Formal Logic
publication date: 2012-11-23T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ndjfl/1348524118
review text: The author considers normal ideals \(I\) on some set \(Z\) and asks when the Boolean algebra \(\mathcal{P}(Z)/I\) is a proper forcing. The question is treated in models that satisfy \(\mathsf{PFA}\) or related axioms. Applying an argument of Woodin, the article gives a nice equivalent formulation of \({\mathsf{PFA}}^{+ \omega_1}\) (\(={\mathsf{PFA}}^{++}\)) (and similar forcing axioms like \({\mathsf{MM}}^{+ \omega_1}\)), called \({\mathsf{PFA}}^{+\text{Diag}}\): For every proper forcing \(\mathbb{P}\), and every large enough regular \(\theta\), there are stationarily many elementary submodels \(M\) of \(H_\theta\) of size \(\omega_1\) for which there is an \((M, \mathbb{P})\)-generic \(G \subset \mathbb{P}\) such that whenever \(\dot{S} \in M\) is a name for a stationary subset of \(\omega_1\), then \(\dot{S}^G\), the evaluation of \(\dot{S}\) under \(G\), is stationary. In Section~3 the author shows that the diagonal reflection principle \(\mathsf{DRP}\), a combinatorial principle introduced by the author in an earlier paper, answers a weaker version of the original question. In fact, \(\mathsf{DRP}\) is equivalent to the existence of stationarily many internally club models of size \(\omega_1\) that force over the nonstationary ideal that every stationary subset of \([\lambda]^\omega\) from the ground model remains stationary in the generic ultrapower. In Section~5, from the existence of a super-2-huge cardinal it is shown that there is a model of \({\mathsf{PFA}}^{+ \omega_1}\) in which there are many ideals whose associated forcings are proper. The dual filters of these ideals concentrate on Chang-type models that witness \({\mathsf{PFA}}^{+\text{Diag}}\). Here a cardinal \(\kappa\) is called super-2-huge if there is an elementary embedding \(j\) of the universe into itself that witnesses both the supercompactness and the 2-hugeness of \(\kappa\). The construction uses the standard way of constructing a model of \({\mathsf{PFA}}^{+ \omega_1}\) due to Baumgartne…
Mathematics Subject Classification ID: 03E57
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 6110279
zbMATH Keywords: forcing axioms
zbMATH Keywords: ideals
zbMATH Keywords: duality theorem
zbMATH Keywords: large cardinals
zbMATH Keywords: proper forcing
zbMATH Keywords: PFA
author: Sean Cox
MaRDI profile type: Publication
OpenAlex ID: W2093527621
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description: scientific article; zbMATH DE number 6110279