Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Forcing consequences of $ PFA$ together with the continuum large


Authors: David Asperó and Miguel Angel Mota
Journal: Trans. Amer. Math. Soc. 367 (2015), 6103-6129
MSC (2010): Primary 03E50, 03E57, 03E35, 03E05
DOI: https://doi.org/10.1090/S0002-9947-2015-06205-9
Published electronically: February 13, 2015
MathSciNet review: 3356931
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using this method we prove that the forcing axiom for the class of all finitely proper posets of size $ \aleph _1$ is compatible with $ 2^{\aleph _{0}}> \aleph _2$. In particular, this answers a question of Moore by showing that $ \mho $ does not follow from this arithmetical assumption.


References [Enhancements On Off] (What's this?)

  • [1] Uri Abraham and James Cummings, More results in polychromatic Ramsey theory, Cent. Eur. J. Math. 10 (2012), no. 3, 1004-1016. MR 2902230, https://doi.org/10.2478/s11533-012-0037-3
  • [2] D. Asperó and M.A. Mota, A generalization of Martin's Axiom. Preprint (2012).
  • [3] James E. Baumgartner, Applications of the proper forcing axiom, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913-959. MR 776640 (86g:03084)
  • [4] M. Bekkali, Topics in set theory, Lecture Notes in Mathematics, vol. 1476, Springer-Verlag, Berlin, 1991. Lebesgue measurability, large cardinals, forcing axioms, rho-functions; Notes on lectures by Stevo Todorčević. MR 1119303 (92m:03070)
  • [5] Keith J. Devlin and Havard Johnsbraten, The Souslin problem, Lecture Notes in Mathematics, Vol. 405, Springer-Verlag, Berlin, 1974. MR 0384542 (52 #5416)
  • [6] 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 (89f:03043), https://doi.org/10.2307/1971415
  • [7] D. H. Fremlin, Consequences of Martin's axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR 780933 (86i:03001)
  • [8] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513 (2004g:03071)
  • [9] István Juhász, A weakening of $ \clubsuit $, with applications to topology, Comment. Math. Univ. Carolin. 29 (1988), no. 4, 767-773. MR 982796 (90d:54005)
  • [10] Piotr Koszmider, On strong chains of uncountable functions, Israel J. Math. 118 (2000), 289-315. MR 1776085 (2001g:03091), https://doi.org/10.1007/BF02803525
  • [11] Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1980. An introduction to independence proofs. MR 597342 (82f:03001)
  • [12] Richard Laver, Random reals and Souslin trees, Proc. Amer. Math. Soc. 100 (1987), no. 3, 531-534. MR 891159 (88g:03068), https://doi.org/10.2307/2046443
  • [13] T. Miyamoto, Club guessing on the least uncountable cardinal and $ \textsc {CH}$, unpublished (2008).
  • [14] Justin Tatch Moore, Set mapping reflection, J. Math. Log. 5 (2005), no. 1, 87-97. MR 2151584 (2006c:03076), https://doi.org/10.1142/S0219061305000407
  • [15] Justin Tatch Moore, A five element basis for the uncountable linear orders, Ann. of Math. (2) 163 (2006), no. 2, 669-688. MR 2199228 (2007d:03085), https://doi.org/10.4007/annals.2006.163.669
  • [16] Justin Tatch Moore, Aronszajn lines and the club filter, J. Symbolic Logic 73 (2008), no. 3, 1029-1035. MR 2444284 (2009e:03086), https://doi.org/10.2178/jsl/1230396763
  • [17] Itay Neeman, Forcing with sequences of models of two types, Notre Dame J. Form. Log. 55 (2014), no. 2, 265-298. MR 3201836, https://doi.org/10.1215/00294527-2420666
  • [18] J. Roitman, Adding a random or a Cohen real: topological consequences and the effect on Martin's axiom, Fund. Math. 103 (1979), no. 1, 47-60. MR 535835 (81h:03098)
  • [19] Saharon Shelah, Independence results, J. Symbolic Logic 45 (1980), no. 3, 563-573. MR 583374 (82b:03099), https://doi.org/10.2307/2273423
  • [20] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982. MR 675955 (84h:03002)
  • [21] Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206 (98m:03002)
  • [22] R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. (2) 94 (1971), 201-245. MR 0294139 (45 #3212)
  • [23] Stevo Todorčević, A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colo., 1983) Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 209-218. MR 763902 (86f:03089), https://doi.org/10.1090/conm/031/763902
  • [24] Stevo Todorčević, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), no. 2, 711-723. MR 792822 (87a:03084), https://doi.org/10.2307/2000309
  • [25] Boban Veličković, Forcing axioms and stationary sets, Adv. Math. 94 (1992), no. 2, 256-284. MR 1174395 (93k:03045), https://doi.org/10.1016/0001-8708(92)90038-M

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03E50, 03E57, 03E35, 03E05

Retrieve articles in all journals with MSC (2010): 03E50, 03E57, 03E35, 03E05


Additional Information

David Asperó
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
Email: d.aspero@uea.ac.uk

Miguel Angel Mota
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email: motagaytan@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2015-06205-9
Received by editor(s): October 4, 2011
Received by editor(s) in revised form: May 31, 2013
Published electronically: February 13, 2015
Additional Notes: The second author was supported by the Austrian Science Fund FWF Project P22430. Both authors were also partially supported by Ministerio de Educación y Ciencia Project MTM2008–03389 (Spain) and by Generalitat de Catalunya Project 2009SGR–00187 (Catalonia).
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society