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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.

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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.
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J. Baumgartner , J. Malitz  and W. Reinhardt , ‘Embedding trees in the rationals’, Proc. Natl Acad. Sci. USA 67 (1970), 17481753.

S. Ben-David  and S. Shelah , ‘Souslin trees and successors of singular cardinals’, Ann. Pure Appl. Logic 30(3) (1986), 207217.

C. C. Chang  and H. J. Keisler , Model Theory, 3rd edn, Studies in Logic and the Foundations of Mathematics, 73 (North-Holland Publishing Co., Amsterdam, 1990).

J. Cummings  and M. Magidor , ‘Martin’s maximum and weak square’, Proc. Amer. Math. Soc. 139(9) (2011), 33393348.

K. J. Devlin , ‘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.

K. J. Devlin  and H. Johnsbrȧten , The Souslin Problem, Lecture Notes in Mathematics, 405 (Springer, Berlin, 1974).

D. H. Fremlin , Consequences of Martin’s Axiom, Cambridge Tracts in Mathematics, 84 (Cambridge University Press, Cambridge, 1984).

R. B. Jensen , ‘The fine structure of the constructible hierarchy’, Ann. Math. Logic 4 229308. erratum, ibid. 4 (1972), 443, 1972. With a section by Jack Silver.

R. Laver  and S. Shelah , ‘The 2 -Souslin hypothesis’, Trans. Amer. Math. Soc. 264(2) (1981), 411417.

A. Rinot , ‘On guessing generalized clubs at the successors of regulars’, Ann. Pure Appl. Logic 162(7) (2011), 566577.

A. Rinot , ‘Chromatic numbers of graphs—large gaps’, Combinatorica 35(2) (2015), 215233.

A. Rinot , ‘Putting a diamond inside the square’, Bull. Lond. Math. Soc. 47(3) (2015), 436442.

A. Shani , ‘Fresh subsets of ultrapowers’, Arch. Math. Logic 55(5) (2016), 835845.

S. Shelah , Proper Forcing, Lecture Notes in Mathematics, 940 (Springer, Berlin–New York, 1982).

S. Shelah  and L. Stanley , ‘Weakly compact cardinals and nonspecial Aronszajn trees’, Proc. Amer. Math. Soc. 104(3) (1988), 887897.

S. Todorčević , ‘Partitioning pairs of countable ordinals’, Acta Math. 159(3–4) (1987), 261294.

S. Todorčević  and V. Torres Pérez , ‘Conjectures of Rado and Chang and special Aronszajn trees’, MLQ Math. Log. Q. 58(4–5) (2012), 342347.

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Forum of Mathematics, Sigma
  • ISSN: -
  • EISSN: 2050-5094
  • URL: /core/journals/forum-of-mathematics-sigma
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