Archive for Mathematical Logic

, Volume 58, Issue 1–2, pp 1–26 | Cite as

Diagonal reflections on squares

  • Gunter FuchsEmail author


The effects of (bounded versions of) the forcing axioms \(\mathsf {SCFA}\), \(\mathsf {PFA}\) and \(\mathsf {MM}\) on the failure of weak threaded square principles of the form \(\square (\lambda ,\kappa )\) are analyzed. To this end, a diagonal reflection principle, \(\mathsf {DSR}{\left( {<}\kappa ,S\right) }\) is introduced. It is shown that \(\mathsf {SCFA} \) implies \(\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }\), for all regular \(\lambda \ge \omega _2\), and that \(\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }\) implies the failure of \(\square (\lambda ,\omega _1)\) if \(\lambda >\omega _2\), and it implies the failure of \(\square (\lambda ,\omega )\) if \(\lambda =\omega _2\). It is also shown that this result is sharp. It is noted that \(\mathsf {MM}\)/\(\mathsf {PFA}\) imply the failure of \(\square (\lambda ,\omega _1)\), for every regular \(\lambda >\omega _1\), and that this result is sharp as well.


Square principles Stationary reflection Forcing axioms Subcomplete forcing Resurrection axioms Continuum hypothesis 

Mathematics Subject Classification

03E50 03E57 03E35 03E55 03E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Claverie, B., Schindler, R.: Woodin’s axiom \((*)\), bounded forcing axioms, and precipitous ideals on \(\omega _1\). J. Symb. Log. 77(2), 475–498 (2012)CrossRefzbMATHGoogle Scholar
  2. 2.
    Cummings, J., Foreman, M., Magidor, M.: Squares, scales and stationary reflection. J. Math. Log. 01(01), 35–98 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cummings, J., Magidor, M.: Martin’s maximum and weak square. Proc. Am. Math. Soc. 139(9), 3339–3348 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Džamonja, M., Hamkins, J.D.: Diamond (on the regulars) can fail at any strongly unfoldable cardinal. Ann. Pure Appl. Log. 144, 83–95 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fuchs, G.: Hierarchies of forcing axioms, the continuum hypothesis and square principles. J. Symb. Log. Preprint available at (2016)
  6. 6.
    Fuchs, G.: Hierarchies of (virtual) resurrection axioms. J. Symb. Log. Preprint available at (2016)
  7. 7.
    Fuchs, G.: Closure properties of parametric subcompleteness. Arch. Math. Log. Preprint available at (2017)
  8. 8.
    Fuchs, G., Rinot, A.: Weak square and stationary reflection. Acta Math. Hung. (2017). Preprint at arXiv:1711.06213 [math.LO]
  9. 9.
    Hamkins, J.D., Johnstone, T.A.: Resurrection axioms and uplifting cardinals. Arch. Math. Log. 53(3–4), 463–485 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hamkins, J.D., Johnstone, T.A.: Strongly uplifting cardinals and the boldface resurrection axioms. Arch. Math. Log. 56(7–8), 1115–1133 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hayut, Y., Lambie-Hanson, C.: Simultaneous stationary reflection and square sequences. J. Math. Log. 17(2) (2017). Preprint: arXiv: 1603.05556v1 [math.LO]
  12. 12.
    Jensen, R.B.: Some remarks on \(\square \) below \(0^{\text{pistol}}\). Circulated notes (unpublished)Google Scholar
  13. 13.
    Jensen, R.B.: Forcing axioms compatible with CH. Handwritten notes, available at (2009)
  14. 14.
    Jensen, R.B.: Subproper and subcomplete forcing. 2009. Handwritten notes, available at
  15. 15.
    Jensen, R.B.: Subcomplete forcing and \({\cal{L}}\)-forcing. In: Chong, C., Feng, Q., Slaman, T.A., Woodin, W.H., Yang, Y. (eds.) E-recursion, forcing and \(C^*\)-algebras, volume 27 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, pp. 83–182. World Scientific, Singapore (2014)Google Scholar
  16. 16.
    Kurepa, G.: Ensembles ordonnés et ramifiés. Publ. Math. Univ. Belgrade 4, 1–138 (1935)zbMATHGoogle Scholar
  17. 17.
    Lambie-Hanson, C.: Squares and narrow systems. J. Symb. Log. 82(3), 834–859 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Larson, P.: Separating stationary reflection principles. J. Symb. Log. 65(1), 247–258 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Laver, R.: Making the supercompactness of \(\kappa \) indestructible under \(\kappa \)-directed closed forcing. Isr. J. Math. 29(4), 385–388 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Magidor, M., Lambie-Hanson, C.: On the strengths and weaknesses of weak squares. In: Appalachian Set Theory 2006–2012, volume 406 of London Mathematical Society Lecture Notes Series, pp. 301–330. Cambridge University Press, Cambridge (2013)Google Scholar
  21. 21.
    Minden, K.: On subcomplete forcing. Ph.D. thesis, The CUNY Graduate Center (2017). Preprint: arXiv:1705.00386 [math.LO]
  22. 22.
    Moore, J.T.: Set mapping reflection. J. Math. Log. 5(1), 87–97 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schimmerling, E.: Coherent sequences and threads. Adv. Math. 216, 89–117 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Todorčević, S.: A note on the proper forcing axiom. Contemp. Math. 31, 209–218 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tsaprounis, K.: On resurrection axioms. J. Symb. Log. 80(2), 587–608 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Veličković, B.: Jensen’s \(\Box \) principles and the Novák number of partially ordered sets. J. Symb. Log. 51(1), 47–58 (1986)CrossRefzbMATHGoogle Scholar
  27. 27.
    Weiß, C.: Subtle and ineffable tree properties. Ph.D. thesis, Ludwig-Maximilians-Universität München (2010)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The College of Staten Island (CUNY)Staten IslandUSA
  2. 2.The Graduate Center (CUNY)New YorkUSA

Personalised recommendations