Abstract
A highly rigid Souslin tree T is constructed such that forcing with T turns T into a Kurepa tree. Club versions of previously known degrees of rigidity are introduced, as follows: for a rigidity property P, a tree T is said to have property P on clubs if for every club set C (containing 0), the restriction of T to levels in C has property P. The relationships between these rigidity properties for Souslin trees are investigated, and some open questions are stated.
Similar content being viewed by others
References
Abraham, U.: Construction of a rigid Aronszajn tree. In: Proceedings of the American Mathematical Society, vol. 77, no. 1, pp. 136–137 (1979)
Abraham U., Shelah S.: Isomorphism types of Aronszajn trees. Israel J. Math. 50, 75–113 (1985)
Devlin, K.J., Johnsbråten, H.: The Souslin Problem. Lecture Notes in Mathematics, vol. 405. Springer, Berlin (1974)
Fuchs G., Hamkins J.D.: Degrees of rigidity for Souslin trees. J. Symb. Log. 74(2), 423–454 (2009)
Gaifman, H., Specker E.P.: Isomorphism types of trees. In: Proceedings of the American Mathematical Society, vol. 15, pp. 1–6 (1964)
Jech T.: Automorphisms of ω 1-trees. Trans. Am. Math. Soc. 173, 57–70 (1972)
Jech T.: Forcing with trees and ordinal definability. Ann. Math. Log. 7, 387–409 (1974)
Jech, T.: Set Theory: The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2003)
Jensen, R.B.: Automorphism properties of Souslin continua. Notices of the American Mathematical Society 16,576, 1969. Abstract #69T-E24
Jin R., Shelah S.: Can a small forcing create Kurepa trees. Ann. Pure Appl. Log. 85, 47–68 (1997)
Todorčević S.: Rigid Aronszajn trees. Publications de l’Institut Mathématique (Beograd), Nouvelle Série 41(27), 259–265 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research for this work was supported by PSC CUNY grant 60048-40-41.
Rights and permissions
About this article
Cite this article
Fuchs, G. Club degrees of rigidity and almost Kurepa trees. Arch. Math. Logic 52, 47–66 (2013). https://doi.org/10.1007/s00153-012-0306-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-012-0306-7