Journal of the Mathematical Society of Japan

Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom

Teruyuki YORIOKA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó–Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó–Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}= {\mathrm{add}}(\mathcal{N}) = 2^{\aleph_0}>\aleph_2$ and ${MA}_{\aleph_1}$ fails.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 913-943.

Dates
First available in Project Euclid: 12 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1499846513

Digital Object Identifier
doi:10.2969/jmsj/06930913

Mathematical Reviews number (MathSciNet)
MR3685031

Zentralblatt MATH identifier
06786984

Subjects
Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory 03E17: Cardinal characteristics of the continuum

Keywords
side condition method $\mho$ weak club guessing sequences cardinal invariants gaps Martin's Axiom

Citation

YORIOKA, Teruyuki. Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom. J. Math. Soc. Japan 69 (2017), no. 3, 913--943. doi:10.2969/jmsj/06930913. https://projecteuclid.org/euclid.jmsj/1499846513


Export citation

References

  • U. Avraham and S. Shelah, Martin's axiom does not imply that every two $\aleph_1$-dense sets of reals are isomorphic, Israel J. Math., 38 (1981), 161–176.
  • U. Abraham and S. Todorčević, Partition properties of $\omega_1$ compatible with CH, Fund. Math., 152 (1997), 165–180.
  • D. Asperó and M. A. Mota, Forcing consequences of PFA together with the continuum large, Trans. Amer. Math. Soc., 367 (2015), 6103–6129.
  • D. Asperó and M. A. Mota, A Generalization of Martin's Axiom, Israel J. Math., 210 (2015), 193–231.
  • T. Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc., 281 (1984), 209–213.
  • T. Bartoszyński, Invariants of Measure and Category, Handbook of set theory, Vols. 1–3, Springer, Dordrecht, 2010, 491–555.
  • T. Bartoszyński and H. Judah, Set theory, On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995.
  • J. Baumgartner, Applications of the proper forcing axiom, Handbook of Set-Theoretic Topology, Chapter 21, pp.,913–959.
  • M. Bell, On the combinatorial principle $P(\mathfrak{c})$, Fund. Math., 114 (1981), 149–157.
  • A. Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory, Vols. 1–3, Springer, Dordrecht, 2010, 395–489.
  • T. Eisworth, J. T. Moore and D. Milovich, Iterated forcing and the Continuum Hypothesis, In: Appalachian set theory 2006–2012, (eds. J. Cummings and E. Schimmerling), London Math. Society, Lecture Notes Series, Cambridge University Press, 2013.
  • T. Jech, Set theory, The third millennium edition, revised and expanded. Springer Monographs in Math., Springer-Verlag, Berlin, 2003.
  • K. Kunen, Set theory, An introduction to independence proofs, Studies in Logic and the Foundations of Math., 102, North-Holland Publishing Co., Amsterdam-New York, 1980.
  • K. Kunen, Set theory, Studies in Logic (London), 34, College Publications, London, 2011.
  • J. T. Moore, A five element basis for the uncountable linear orders, Ann. of Math. (2), 163 (2006), 669–688.
  • J. T. Moore, Aronszajn lines and the club filter, J. Symbolic Logic, 73 (2008), 1029–1035.
  • J. T. Moore and D. Milovich, A tutorial on Set Mapping Reflection, In: Appalachian set theory 2006–2012, (eds. J. Cummings and E. Schimmerling), London Math. Society, Lecture Notes Series, Cambridge University Press, 2013.
  • S. Shelah, Proper and improper forcing, Second edition, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.
  • M. Scheepers, Gaps in $\omega^\omega$, In: Set Theory of the Reals, Israel Math. Conference Proceedings, 6 1993, 439–561.
  • R. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. (2), 94 (1971), 201–245.
  • S. Todorčević, A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984, 209–218.
  • S. Todorčević, Directed sets and cofinal types, Transactions of American Math. Society, 290, no. 2, 1985, pp.,711–723.
  • S. Todorčević, Partition Problems in Topology, Contemporary mathematics, 84, American Math. Society, Providence, Rhode Island, 1989.
  • S. Todorčević, S-gaps and T-gaps, Handwritten note, August 6, 2005.
  • S. Todorčević and I. Farah, Some applications of the method of forcing, Yenisei Series in Pure and Applied Math., Yenisei, Moscow; Lycée, Troitsk, 1995.
  • S. Todorčević and B. Veličković, Martin's axiom and partitions, Compositio Math., 63 (1987), 391–408.
  • T. Yorioka, The diamond principle for the uniformity of the meager ideal implies the existence of a destructible gap, Arch. Math. Logic, 44 (2005), 677–683.
  • T. Yorioka, Combinatorial principles on $\omega_1$, cardinal invariants of the meager ideal and destructible gaps, J. Math. Soc. Japan, 57 (2005), 1217–1228.
  • T. Yorioka, Some weak fragments of Martin's axiom related to the rectangle refining property, Arch. Math. Logic, 47 (2008), 79–90.
  • T. Yorioka, A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees, Ann. Pure Appl. Logic, 161 (2010), 469–487.