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PFA and ideals on \(\omega_{2}\) whose associated forcings are proper. (English) Zbl 1253.03078

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 Baumgartner, utilizing an adaptation of Laver functions to huge embeddings.
In the last section, the author achieves an even stronger conclusion from the existence of a super-3-huge cardinal. The ideals of the previous theorem are not only proper, they are projections of higher ideals such that these projections are also forcing projections.
Reviewer’s remark: The author informed the reviewer of a mistake in the last paragraph of Section 4. In there, it is claimed that \(\text{FP}(F(U), F(\text{proj}(U)))\) holds in \(V[G]\). While (1)–(3) give a forcing projection \(\mathbb{P}_{F(U)} \to \mathbb{P}_{F(\text{proj}(U))}\), this is not necessarily the canonical ideal projection map. In fact it is consistent that \(\text{FP}(F(U), F(\text{proj}(U)))\) fails in \(V^{\text{Coll}(\omega_1, <\!\kappa)}\) for some \(\kappa\).
This does not affect Section 6, the map \(k\) there is the identity on the relevant objects and the given proof correct. Also, Section 4 does correctly show that \(\mathsf{MA}^{+\text{Diag}}(\mathbb{P}_{F(\text{proj}(U))})\) holds.

MSC:

03E57 Generic absoluteness and forcing axioms
03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
03E55 Large cardinals
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References:

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