#paper
# Abstract
In [[Schlindwein - Consistency of Suslin's hypothesis, a nonspecial Aronszajn tree, and GCH]], the consistency of "ZFC + continuum hypothesis + Suslin's hypothesis + not Every Aronszajn Tree is Special (~EATS)" is demonstrated using proper forcing. In fact, the following stronger statement is shown consistent:
$(*)$ ZFC + CH + there is an Aronszajn tree $T^*$ and a stationary co-stationary set $S^*$ such that $T^*$ is S-st-special iff $S - S^*$ is non-stationary, and every Aronszajn tree is $S^*$-st-special.
See [Sh, p. 286] for the definition of S-st-special. The proof relies heavily on [Sh, chap. V].
Of course the consistency of CH with SH is due to Jensen [DJ]. In this paper, we build a model of
$(*)$ + every $\omega_{1}$-tree is $S^{*}$-$*$-special (and therefore there are no Suslin or Kurepa trees)
starting from an inaccessible cardinal.
Consequently, every combination of ZFC ± CH ± SH ± KH ± EATS not ruled out by EATS==>SH, is attained.
The main new ingredient of the proof, compared to Sch2, is the use of a method of Todorcevic. Familiarity with Sch2 is not assumed, but familiarity with Sh, chaps. III & V is.