Coherent forests
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- Proc. Amer. Math. Soc. 143 (2015), 2705-2717 Request permission
Abstract:
A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be constructed within ZFC. We give several ways of obtaining coherent Suslin forests by forcing, one of which generalizes the well-known argument of Todorčević that a Cohen real adds a Suslin tree. Another uses a strong combinatorial principle that plays a similar role to diamond. We show that, starting from a large cardinal, this principle can be obtained by a forcing that is small relative to the forest it constructs.References
- Uri Abraham and Stevo Todorčević, Partition properties of $\omega _1$ compatible with CH, Fund. Math. 152 (1997), no. 2, 165–181. MR 1441232, DOI 10.4064/fm-152-2-165-181
- Eskew, Monroe, Measurability properties on small cardinals. Ph.D. Thesis, UC Irvine, 2014.
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Thomas J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1972/73), 165–198. MR 325397, DOI 10.1016/0003-4843(73)90014-4
- Piotr Koszmider, On coherent families of finite-to-one functions, J. Symbolic Logic 58 (1993), no. 1, 128–138. MR 1217181, DOI 10.2307/2275329
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- Stevo Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261–294. MR 908147, DOI 10.1007/BF02392561
- Stevo Todorčević, A dichotomy for P-ideals of countable sets, Fund. Math. 166 (2000), no. 3, 251–267. MR 1809418, DOI 10.4064/fm-166-3-251-267
- Weiß, Christoph. Subtle and ineffable tree properties. Ph.D. Thesis, Ludwig Maximilians Universität München, 2010.
Additional Information
- Monroe Eskew
- Affiliation: International Research Fellow of the Japan Society for the Promotion of Science. Department of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan
- MR Author ID: 1101378
- ORCID: 0000-0001-8094-9731
- Received by editor(s): November 21, 2013
- Published electronically: February 4, 2015
- Communicated by: Mirna Dzamonja
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2705-2717
- MSC (2010): Primary 03E05, 03E10, 03E20, 03E35, 03E55
- DOI: https://doi.org/10.1090/S0002-9939-2015-12622-6
- MathSciNet review: 3326048