#paper #tree #souslin #SH #aronszajn # Abstract Shelah has shown that Suslin’s hypothesis does not imply every Aronszajn tree is special. We improve this result by constructing a model of Suslin’s hypothesis in which some Aronszajn tree has no antichain whose levels constitute a stationary set. The main point is a new preservation theorem, the proof of which illustrates the usefulness of certain ideas in [8, Section 11. # Main **Main Theorem** It is consistent that [[Trees#^dde5ed|Souslin's hypothesis]] holds and there is an [[Trees#^almost-souslin|almost souslin]] tree. *Proof idea.* Start with a model of GCH where $T^{*}$ is a Souslin tree. Form a forcing iteration of length $\omega_{2}$ where each step either kills the Souslinity of some tree, or the stationarity of some antichain in $T^{*}$. The forcings used are proper and will be "$T^{*}$-strongly preserving", the final iteration has $\aleph_{2}$-c.c. ## Preservation of Aronszajn ![[Pasted image 20230423094649.png]] ![[Pasted image 20230423094954.png]] ![[Pasted image 20230423095006.png]] ![[Pasted image 20230423095247.png]] ## Shooting clubs ![[Pasted image 20230423102838.png]] ![[Pasted image 20230423102939.png]] ![[Pasted image 20230423103621.png]] ![[Pasted image 20230423103959.png]] ![[Pasted image 20230423104032.png]] ![[Pasted image 20230423103911.png]] This is proved in section 2. The definition is Def 31, but is quite complicated and relies on previous definitions.