##
**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.

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.

Reviewer: Christoph Weiss (Irvine)

### 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 |

PDFBibTeX
XMLCite

\textit{S. Cox}, Notre Dame J. Formal Logic 53, No. 3, 397--412 (2012; Zbl 1253.03078)

### References:

[1] | Beaudoin, R. E., “The proper forcing axiom and stationary set reflection,” Pacific Journal of Mathematics , vol. 149 (1991), pp. 13-24. · Zbl 0687.03034 · doi:10.2140/pjm.1991.149.13 |

[2] | Corazza, P., “Laver sequences for extendible and super-almost-huge cardinals,” Journal of Symbolic Logic , vol. 64 (1999), pp. 963-983. · Zbl 0949.03046 · doi:10.2307/2586614 |

[3] | Cox, S., “Ideal projections as forcing projections,” in preparation. · Zbl 1353.03059 |

[4] | Cox, S., “The Diagonal Reflection Principle,” Proceedings of the American Mathematical Society , vol. 140 (2012), pp. 2893-2902. · Zbl 1291.03082 · doi:10.1090/S0002-9939-2011-11103-1 |

[5] | Cummings, J., “Iterated forcing and elementary embeddings,” pp. 775-883 in Handbook of Set Theory , vol. 2, edited by M. Foreman and A. Konamori, Springer, Dordrecht, 2010. · Zbl 1198.03060 · doi:10.1007/978-1-4020-5764-9_13 |

[6] | Foreman, M., “Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals,” Advances in Mathematics , vol. 222 (2009), pp. 565-595. · Zbl 1178.03068 · doi:10.1016/j.aim.2009.05.006 |

[7] | Foreman, M., “Ideals and generic elementary embeddings,” pp. 885-1147 in Handbook of Set Theory , Springer, Dordrecht, 2010. · Zbl 1198.03050 · doi:10.1007/978-1-4020-5764-9_14 |

[8] | Foreman, M., “Calculating quotient algebras for generic embeddings,” to appear in Israel Journal of Mathematics . · Zbl 1270.03092 |

[9] | Foreman, M., and M. Magidor, “Large cardinals and definable counterexamples to the continuum hypothesis,” Annals of Pure and Applied Logic , vol. 76 (1995), pp. 47-97. · Zbl 0837.03040 · doi:10.1016/0168-0072(94)00031-W |

[10] | Foreman, M., M. Magidor, and S. Shelah, “Martin’s maximum, saturated ideals, and nonregular ultrafilters, I,” Annals of Mathematics (2) , vol. 127 (1988), pp. 1-47. · Zbl 0645.03028 · doi:10.2307/1971415 |

[11] | Jech, T., Set Theory , 3rd millennium edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. |

[12] | Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings , 2nd edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. · Zbl 1022.03033 |

[13] | Viale, M., “Guessing models and generalized Laver diamond,” Annals of Pure and Applied Logic , published electronically 29 December 2011, . · Zbl 1270.03084 · doi:10.1016/j.apal.2011.12.015 |

[14] | Woodin, W. H., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal , vol. 1 of de Gruyter Series in Logic and its Applications , Walter de Gruyter & Co., Berlin, 1999. · Zbl 0954.03046 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.