Strongly proper forcing and some problems of Foreman
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- by Sean Cox and Monroe Eskew PDF
- Trans. Amer. Math. Soc. 371 (2019), 5039-5068 Request permission
Abstract:
We answer several questions of Foreman, most of which are closely related to Mitchell’s notion of strongly proper forcing. We prove that presaturation of a normal ideal implies projective antichain catching, providing a solution to a problem of Foreman about ideal projections that is more comprehensive and simpler than the earlier solution obtained by Cox and Zeman. We answer an older question of Foreman about the relationship between generic hugeness and generic almost hugeness. Finally, we answer two technical questions of Foreman related to his Duality Theorem.References
- Uri Abraham and Saharon Shelah, Forcing closed unbounded sets, J. Symbolic Logic 48 (1983), no. 3, 643–657. MR 716625, DOI 10.2307/2273456
- James E. Baumgartner and Alan D. Taylor, Saturation properties of ideals in generic extensions. II, Trans. Amer. Math. Soc. 271 (1982), no. 2, 587–609. MR 654852, DOI 10.1090/S0002-9947-1982-0654852-4
- Douglas R. Burke, Precipitous towers of normal filters, J. Symbolic Logic 62 (1997), no. 3, 741–754. MR 1472122, DOI 10.2307/2275571
- Sean Cox and Martin Zeman, Ideal projections and forcing projections, J. Symb. Log. 79 (2014), no. 4, 1247–1285. MR 3343538, DOI 10.1017/jsl.2013.24
- James Cummings, Iterated forcing and elementary embeddings, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 775–883. MR 2768691, DOI 10.1007/978-1-4020-5764-9_{1}3
- Qi Feng and Thomas Jech, Projective stationary sets and a strong reflection principle, J. London Math. Soc. (2) 58 (1998), no. 2, 271–283. MR 1668171, DOI 10.1112/S0024610798006462
- Matthew Foreman, Calculating quotient algebras of generic embeddings, Israel J. Math. 193 (2013), no. 1, 309–341. MR 3038554, DOI 10.1007/s11856-012-0118-9
- Matthew Foreman, Ideals and generic elementary embeddings, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 885–1147. MR 2768692, DOI 10.1007/978-1-4020-5764-9_{1}4
- Matthew Foreman, Potent axioms, Trans. Amer. Math. Soc. 294 (1986), no. 1, 1–28. MR 819932, DOI 10.1090/S0002-9947-1986-0819932-2
- Matthew Foreman and Menachem Magidor, Large cardinals and definable counterexamples to the continuum hypothesis, Ann. Pure Appl. Logic 76 (1995), no. 1, 47–97. MR 1359154, DOI 10.1016/0168-0072(94)00031-W
- M. Foreman, M. Magidor, and S. Shelah, Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2) 127 (1988), no. 1, 1–47. MR 924672, DOI 10.2307/1971415
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, DOI 10.2307/2271949
- Menachem Magidor, On the existence of nonregular ultrafilters and the cardinality of ultrapowers, Trans. Amer. Math. Soc. 249 (1979), no. 1, 97–111. MR 526312, DOI 10.1090/S0002-9947-1979-0526312-2
- William J. Mitchell, On the Hamkins approximation property, Ann. Pure Appl. Logic 144 (2006), no. 1-3, 126–129. MR 2279659, DOI 10.1016/j.apal.2006.05.005
- W. Hugh Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, Second revised edition, De Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010. MR 2723878, DOI 10.1515/9783110213171
- Martin Zeman, Two upper bounds on consistency strength of $\neg \square _{\aleph _\omega }$ and stationary set reflection at two successive $\aleph _n$, Notre Dame J. Form. Log. 58 (2017), no. 3, 409–432. MR 3681102, DOI 10.1215/00294527-2017-0005
Additional Information
- Sean Cox
- Affiliation: Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, Virginia 23284
- MR Author ID: 883409
- Email: scox9@vcu.edu
- Monroe Eskew
- Affiliation: Kurt Gödel Research Center, University of Vienna, Währinger Strasse 25, 1090 Wien, Austria
- MR Author ID: 1101378
- ORCID: 0000-0001-8094-9731
- Email: monroe.eskew@univie.ac.at
- Received by editor(s): December 5, 2016
- Received by editor(s) in revised form: March 12, 2018
- Published electronically: December 28, 2018
- Additional Notes: The first author acknowledges support from Simons Grant No. 318467.
Both authors gratefully acknowledge support from the VCU Presidential Research Quest Fund. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5039-5068
- MSC (2010): Primary 03E05, 03E35, 03E55, 03E57, 03E65
- DOI: https://doi.org/10.1090/tran/7725
- MathSciNet review: 3934477