Skip to main content
Springer Nature Link
Log in

Two applications of finite side conditions at \(\omega _2\)

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We present two applications of forcing with finite sequences of models as side conditions, adding objects of size \(\omega _2\). The first involves adding a \(\Box _{\omega _1}\) sequence and variants of such sequences. The second involves adding partial weak specializing functions for trees of height \(\omega _2\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asperó, D., Mota, M.A.: A generalization of Martin’s axiom. Isr. J. Math. 210, 193–231 (2015). doi:10.1007/s11856-015-1250-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Asperó, D., Mota, M.A.: Forcing consequences of \(PFA\) together with the continuum large. Trans. Am. Math. Soc. 367(9), 6103–6129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baumgartner, J.E.: Applications of the proper forcing axiom. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, pp. 913–959. North-Holland, Amsterdam (1984)

  4. Baumgartner, J.E., Shelah, S.: Remarks on superatomic Boolean algebras. Ann. Pure Appl. Logic 33(2), 109–129 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cummings, J.: Souslin trees which are hard to specialise. Proc. Am. Math. Soc. 125(8), 2435–2441 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dolinar, G., Džamonja, M.: Forcing \(\square _{\omega _1}\) with finite conditions. Ann. Pure Appl. Logic 164(1), 49–64 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Friedman, S.-D.: Forcing with finite conditions. In: Set Theory, Trends Mathematics, pp. 285–295. Birkhäuser, Basel (2006)

  8. Friedman, S.-D.: BPFA and inner models. Ann. Jpn. Assoc. Philos. Sci. 19, 29–36 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Friedman, S.-D., Krueger, J.: Thin stationary sets and disjoint club sequences. Trans. Am. Math. Soc. 359(5), 2407–2420 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koszmider, P.: On strong chains of uncountable functions. Isr. J. Math. 118, 289–315 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krueger, J.: Coherent adequate sets and forcing square. Fund. Math. 224(3), 279–300 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Krueger, J., Mota, M.A.: Coherent adequate forcing and preserving CH. J. Math. Logic 15(2), 1550005 (2015). doi:10.1142/S0219061315500051

    Article  MathSciNet  MATH  Google Scholar 

  13. Mitchell, W.J.: Adding closed unbounded subsets of \(\omega _2\) with finite forcing. Notre Dame J. Formal Logic 46(3), 357–371 (2005)

    Article  MathSciNet  Google Scholar 

  14. Mitchell, W.J.: \(I[\omega _2]\) can be the nonstationary ideal on \({\rm Cof}(\omega _1)\). Trans. Am. Math. Soc. 361(2), 561–601 (2009)

    Article  MATH  Google Scholar 

  15. Neeman, I.: Higher analogues of the proper forcing axiom (in preparation)

  16. Neeman, I.: Forcing with sequences of models of two types. Notre Dame J. Form. Log. 55(2), 265–298 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rubin, M., Shelah, S.: Combinatorial problems on trees: partitions, \(\Delta \)-systems and large free subtrees. Ann. Pure Appl. Logic 33(1), 43–81 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sakai, H.: Chang’s conjecture and weak square. Arch. Math. Logic 52(1–2), 29–45 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shelah, S.: Proper Forcing. Lecture Notes in Mathematics, vol. 940. Springer, Berlin (1982)

  20. Shelah, S., Stanley, L.: Weakly compact cardinals and nonspecial Aronszajn trees. Proc. Am. Math. Soc. 104(3), 887–897 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Todorčević, S.: Stationary sets, trees and continuums. Publ. Inst. Math. (Beograd) (N.S.) 29(43), 249–262 (1981)

    MathSciNet  MATH  Google Scholar 

  22. Stevo, T.: A note on the proper forcing axiom. In: Axiomatic set theory (Boulder, CO, 1983), volume 31 of Contemporary Mathematics, pp. 209–218. American Mathematical Society, Providence, RI (1984)

  23. Todorčević, S.: Directed sets and cofinal types. Trans. Am. Math. Soc. 290(2), 711–723 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Todorčević, S.: Partition relations for partially ordered sets. Acta Math. 155(1–2), 1–25 (1985)

    MathSciNet  MATH  Google Scholar 

  25. Todorčević, S.: Special square sequences. Proc. Am. Math. Soc. 105(1), 199–205 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stevo, T.: Conjectures of Rado and Chang and cardinal arithmetic. In: Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991), volume 411 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pp. 385–398. Kluwer Academic Publications, Dordrecht (1993)

  27. Velickovic, B., Venturi, G.: Proper forcing remastered. In: Cummings, J., Schimmerling, E. (eds.) Appalachian Set Theory: 2006–2012, London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2013)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California Los Angeles, Los Angeles, CA, 90095-1555, USA

    Itay Neeman

Authors
  1. Itay Neeman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Itay Neeman.

Additional information

This material is based upon work supported by the National Science Foundation under Grants No. DMS-1101204 and DMS-1363364, and the Simons Foundation under Simons Fellowship No. 225854.

The material from Definition 3.26 to the end of Section 3 was added in revision in July 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Neeman, I. Two applications of finite side conditions at \(\omega _2\) . Arch. Math. Logic 56, 983–1036 (2017). https://doi.org/10.1007/s00153-017-0550-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-017-0550-y

Keywords

Mathematics Subject Classification

Search

Navigation

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy