Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T01:46:50.796Z Has data issue: false hasContentIssue false
  • English
  • Français

SQUARE WITH BUILT-IN DIAMOND-PLUS

Published online by Cambridge University Press:  08 September 2017

ASSAF RINOT  and
RALF SCHINDLER
ASSAF RINOT
Affiliation:
DEPARTMENT OF MATHEMATICS BAR-ILAN UNIVERSITY RAMAT-GAN52900, ISRAELE-mail: [email protected]: http://www.assafrinot.com
RALF SCHINDLER
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG UNIVERSITÄT MÜNSTER, EINSTEINSTR. 62 48149 MÜNSTER, GERMANYE-mail: [email protected]: http://www.math.uni-muenster.de/logik/personen/rds
Get access Rights & Permissions [Opens in a new window]

Abstract

We formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds in L for every infinite cardinal.

As an application, we prove that the following two hold in L:

  1. 1. For every infinite regular cardinal λ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;

  2. 2. For every infinite cardinal λ, there exists a respecting λ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed in L.

Keywords

Primary 03E45Secondary 03E05diamond principlesquare principleconstructibilitywalks on ordinalsKurepa treealmost Souslin treeparameterized proxy principle
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 809 - 833
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abraham, U., Shelah, S., and Solovay, R. M., Squares with diamonds and Souslin trees with special squares . Fundamenta Mathematicae, vol. 127 (1987), no. 2, pp. 133–162.Google Scholar
Brodsky, A. M. and Rinot, A., A microscopic approach to Souslin-tree constructions. Part I . Annals of Pure and Applied Logic, doi:10.1016.j.apal.2017.05.003.Google Scholar
Brodsky, A. M. and Rinot, A., Reduced powers of Souslin trees . Forum of Mathematics, Sigma, vol. 5 (2017), pp. 1–82.Google Scholar
Brodsky, A. M. and Rinot, A., A microscopic approach to Souslin-tree constructions. Part II, in preparation, 2017.Google Scholar
Devlin, K. J., The combinatorial principle ⋄♯, this Journal, vol. 47 (1982), no. 4, pp. 888–899.Google Scholar
Devlin, K. J., Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984.Google Scholar
Devlin, K. J. and Shelah, S., Souslin properties and tree topologies . Proceedings of the London Mathematical Society, Third Series, vol. 39 (1979), no. 2, pp. 237–252.Google Scholar
Foreman, M. and Komjath, P., The club guessing ideal: Commentary on a theorem of Gitik and Shelah . Journal of Mathematical Logic, vol. 5 (2005), no. 1, pp. 99–147.Google Scholar
Gray, C. W., Iterated forcing from the strategic point of view, Ph.D. thesis, ProQuest LLC, Ann Arbor, MI, University of California, Berkeley, 1980.Google Scholar
Hrušák, M. and Martínez Ranero, C., Some remarks on non-special coherent Aronszajn trees . Acta Universitatis Carolinae. Mathematica et Physica, vol. 46 (2005), no. 2, pp. 33–40.Google Scholar
Ishiu, T., Club guessing sequences and filters, this Journal, vol. 70 (2005), no. 4, pp. 1037–1071.Google Scholar
Ishiu, T. and Larson, P. B., Some results about (+) proved by iterated forcing, this Journal, vol. 77 (2012), no. 2, pp. 515–531.Google Scholar
Jensen, R. B., The fine structure of the constructible hierarchy . Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308; erratum, ibid. 4(1972), 443.Google Scholar
Jensen, R., Schimmerling, E., Schindler, R., and Steel, J., Stacking mice, this Journal, vol. 74 (2009), no. 1, pp. 315–335.Google Scholar
Rinot, A., Chain conditions of products, and weakly compact cardinals . Bulletin of Symbolic Logic, vol. 20 (2014), no. 3, pp. 293–314.Google Scholar
Rinot, A., Putting a diamond inside the square . Bulletin of the London Mathematical Society, vol. 47 (2015), no. 3, pp. 436–442.Google Scholar
Rinot, A., Hedetniemi’s conjecture for uncountable graphs . Journal of the European Mathematical Society, vol. 19 (2017), no. 1, pp. 285–298.Google Scholar
Schindler, R., Set theory, Exploring Independence and Truth, Universitext, Springer, Cham, 2014.Google Scholar
Steinhorn, C. I. and King, J. H., The uniformization property for ℵ2 . Israel Journal of Mathematics, vol. 36 (1980), no. 3–4, pp. 248–256.Google Scholar
Todorčević, S. B., Trees, subtrees and order types . Annals of Mathematical Logic, vol. 20 (1981), no. 3, pp. 233–268.Google Scholar
Todorčević, S. B., Partitioning pairs of countable ordinals . Acta Mathematica, vol. 159 (1987), no. 3–4, pp. 261–294.Google Scholar
Todorčević, S. B., Representing trees as relatively compact subsets of the first Baire class . Bulletin. Classe de Sciences, Mathématiques et Naturelles. Sciences Mathématiques, vol. 131 (2005), no. 30, pp. 29–45.Google Scholar
Todorčević, S. B., Walks on Ordinals and their Characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007.Google Scholar