Skip to main content
×
Home

REDUCED POWERS OF SOUSLIN TREES

  • ARI MEIR BRODSKY (a1) and ASSAF RINOT (a2)
Abstract

We study the relationship between a $\unicode[STIX]{x1D705}$ -Souslin tree $T$ and its reduced powers $T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$ .

Previous works addressed this problem from the viewpoint of a single power $\unicode[STIX]{x1D703}$ , whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph _{6}$ -Souslin tree $T$ and a sequence of uniform ultrafilters $\langle {\mathcal{U}}_{n}\mid n<6\rangle$ such that $T^{\aleph _{n}}/{\mathcal{U}}_{n}$ is $\aleph _{6}$ -Aronszajn if and only if $n<6$ is not a prime number.

This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      REDUCED POWERS OF SOUSLIN TREES
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      REDUCED POWERS OF SOUSLIN TREES
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      REDUCED POWERS OF SOUSLIN TREES
      Available formats
      ×
Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
Hide All
[BMR70] Baumgartner J., Malitz J. and Reinhardt W., ‘Embedding trees in the rationals’, Proc. Natl Acad. Sci. USA 67 (1970), 17481753.
[BDS86] Ben-David S. and Shelah S., ‘Souslin trees and successors of singular cardinals’, Ann. Pure Appl. Logic 30(3) (1986), 207217.
[BR15] Brodsky A. M. and Rinot A., ‘A microscopic approach to Souslin-tree constructions. Part I’, Preprint, 2015, arXiv:1601.01821, http://www.assafrinot.com/paper/22.
[BR16] Brodsky A. M. and Rinot A., ‘More notions of forcing add a Souslin tree’, Preprint, 2016, arXiv:1607.07033, http://www.assafrinot.com/paper/26.
[BR17] Brodsky A. M. and Rinot A., ‘A microscopic approach to Souslin-tree constructions. Part II’, in preparation, 2017, http://www.assafrinot.com/paper/23.
[CK90] Chang C. C. and Keisler H. J., Model Theory, 3rd edn, Studies in Logic and the Foundations of Mathematics, 73 (North-Holland Publishing Co., Amsterdam, 1990).
[Cum97] Cummings J., ‘Souslin trees which are hard to specialise’, Proc. Amer. Math. Soc. 125(8) (1997), 24352441.
[CM11] Cummings J. and Magidor M., ‘Martin’s maximum and weak square’, Proc. Amer. Math. Soc. 139(9) (2011), 33393348.
[Dev81] Devlin K. J., ‘Morass-like constructions of 2 -trees in L ’, inSet Theory and Model Theory (Bonn, 1979), Lecture Notes in Mathematics, 872 (Springer, Berlin–New York, 1981), 136.
[Dev83] Devlin K. J., ‘Reduced powers of 2 -trees’, Fund. Math. 118(2) (1983), 129134.
[DJ74] Devlin K. J. and Johnsbrȧten H., The Souslin Problem, Lecture Notes in Mathematics, 405 (Springer, Berlin, 1974).
[Fre84] Fremlin D. H., Consequences of Martin’s Axiom, Cambridge Tracts in Mathematics, 84 (Cambridge University Press, Cambridge, 1984).
[Jen72] Jensen R. B., ‘The fine structure of the constructible hierarchy’, Ann. Math. Logic 4 229308. erratum, ibid. 4 (1972), 443, 1972. With a section by Jack Silver.
[Kan03] Kanamori A., ‘Large cardinals in set theory from their beginnings’, inThe Higher Infinite, 2nd edn, Springer Monographs in Mathematics (Springer, Berlin, 2003).
[Kur52] Kurepa D., ‘Sur une propriété caractéristique du continu linéaire et le problème de Suslin’, Acad. Serbe Sci. Publ. Inst. Math. 4 (1952), 97108.
[LH17] Lambie-Hanson C., ‘Aronszajn trees, square principles, and stationary reflection’, MLQ Math. Log. Q., to appear.
[LS81] Laver R. and Shelah S., ‘The 2 -Souslin hypothesis’, Trans. Amer. Math. Soc. 264(2) (1981), 411417.
[Luc17] Lücke P., ‘Ascending paths and forcings that specialize higher Aronszajn trees’, Fund. Math. (2017), to appear.
[Rin11] Rinot A., ‘On guessing generalized clubs at the successors of regulars’, Ann. Pure Appl. Logic 162(7) (2011), 566577.
[Rin15a] Rinot A., ‘Chromatic numbers of graphs—large gaps’, Combinatorica 35(2) (2015), 215233.
[Rin15b] Rinot A., ‘Putting a diamond inside the square’, Bull. Lond. Math. Soc. 47(3) (2015), 436442.
[RS17] Rinot A. and Schindler R., ‘Square with built-in diamond-plus’, J. Symbolic Logic (2017), to appear, http://www.assafrinot.com/paper/21.
[Sha16] Shani A., ‘Fresh subsets of ultrapowers’, Arch. Math. Logic 55(5) (2016), 835845.
[She82] Shelah S., Proper Forcing, Lecture Notes in Mathematics, 940 (Springer, Berlin–New York, 1982).
[SS88] Shelah S. and Stanley L., ‘Weakly compact cardinals and nonspecial Aronszajn trees’, Proc. Amer. Math. Soc. 104(3) (1988), 887897.
[Tod87] Todorčević S., ‘Partitioning pairs of countable ordinals’, Acta Math. 159(3–4) (1987), 261294.
[TTP12] Todorčević S. and Torres Pérez V., ‘Conjectures of Rado and Chang and special Aronszajn trees’, MLQ Math. Log. Q. 58(4–5) (2012), 342347.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Forum of Mathematics, Sigma
  • ISSN: -
  • EISSN: 2050-5094
  • URL: /core/journals/forum-of-mathematics-sigma
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 81 *
Loading metrics...

Abstract views

Total abstract views: 188 *
Loading metrics...

* Views captured on Cambridge Core between 16th January 2017 - 22nd November 2017. This data will be updated every 24 hours.