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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Automorphisms of $ \omega \sb{1}$-trees


Author: Thomas J. Jech
Journal: Trans. Amer. Math. Soc. 173 (1972), 57-70
MSC: Primary 02K30
MathSciNet review: 0347605
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Abstract: The number of automorphisms of a normal $ {\omega _1}$-tree $ T$, denoted by $ \sigma (T)$, is either finite or $ {2^{{\aleph _0}}} \leqslant \sigma (T) \leqslant {2^{{\aleph _1}}}$. Moreover, if $ \sigma (T)$ is infinite then $ \sigma {(T)^{{\aleph _0}}} = \sigma (T)$. Moreover, if $ T$ has no Suslin subtree then $ \sigma (T)$ is finite or $ \sigma (T) = {2^{{\aleph _0}}}$ or $ \sigma (T) = {2^{{\aleph _1}}}$. It is consistent that there is a Suslin tree with arbitrary precribed $ \sigma (T)$ between $ {2^{{\aleph _0}}}$ and $ {2^{{\aleph _1}}}$, subject to the restriction above; e.g. $ {2^{{\aleph _0}}} = {\aleph _1},{2^{{\aleph _1}}} = {\aleph _{324}}$ and $ \sigma (T) = {\aleph _{17}}$. We prove related results for Kurepa trees and isomorphism types of trees. We use Cohen's method of forcing and Jensen's techniques in $ L$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0347605-1
PII: S 0002-9947(1972)0347605-1
Keywords: Normal $ {\omega _1}$-tree, Suslin tree, Suslin continuum, Kurepa tree, rigid tree
Article copyright: © Copyright 1972 American Mathematical Society