Abstract
AbstractWe show it is consistent that there is a Souslin tree S such that after forcing with S, S is Kurepa and for all clubs $$C \subset \omega _1$$
C
⊂
ω
1
, $$S\upharpoonright C$$
S
↾
C
is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with $$\diamondsuit $$
♢
that for every Souslin tree T there is a dense $$X \subseteq T$$
X
⊆
T
which does not contain a copy of T. This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982).
Funder
Westfälische Wilhelms-Universität Münster
Publisher
Springer Science and Business Media LLC
Reference4 articles.
1. Baumgartner, J.E.: Order types of real numbers and other uncountable orderings. In: Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., pp 239–277. Reidel, Dordrecht-Boston, Mass (1982)
2. Fuchs, G.: Club degrees of rigidity and almost Kurepa trees. Arch. Math. Logic 52(1–2), 47–66 (2013)
3. Fuchs, G., Hamkins, J.D.: Degrees of rigidity for Souslin trees. J. Symb. Logic 74(2), 423–454 (2009)
4. Kunen, K.: An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1983)