#paper #SH #aronszajn #special-tree #forcing #GCH
# Abstract
Introduction. In [Sh, Chapter IX], Shelah constructs a model of set theory in which Suslin's hypothesis is true, yet there is an Aronszajn tree which is not special. In his model, we have /. He asks whether the same result could be obtained consistently with CH. In this paper, we answer his question in the affirmative.
Let us say that a tree T is S-st-special iff there is a function ƒ with dom(f) = {t ∈ T: rank(t) ∈ S} and for every t1 < t2 both in dom(t) we have f(t2) ≠ f(t1) < rank(t1). In Shelah's model, every tree is S-st-special for some fixed stationary costationary set S. Also, there is some tree T such that T is not S′-st-special whenever S′ – S is stationary. These properties, which are sufficient to ensure that Suslin's hypothesis holds and that T is not special, also hold in the model constructed in this paper. These properties also ensure that every Aronszajn tree has a stationary antichain (i.e., an antichain A such that {rank(t): t ∈ A} is stationary). Hence, it is natural to ask whether there is a model of Suslin's hypothesis in which some Aronszajn tree has no stationary antichain. We answer in the affirmative in [S].The construction we use owes much to Shelah's approach to the theorem, due to Jensen (see [DJ]), that CH is consistent with Suslin's hypothesis. This is given in [Sh, Chapter V].
# Summary
DEFINITION 1. For $S$ a set of countable limit ordinals and $T$ an Aronszajn tree, we say that $T$ is $S$-st-special if there is a function $f$ mapping $T\restriction S$ into $\omega_{1}$ such that whenever $rk(t) \in S$ we have $f (t) < rk(t)$, and if $t <_T s$ are both in $dom(f)$ then $f (t) \ne f (s)$. Here $rk(t)$ is the rank of $t$ in the tree $T$.
## Main theorem
THEOREM 45. If ZFC is consistent, then so is ZFC+GCH+there is a stationary co-stationary set $S^*$ such that every Aronszajn tree is $S^*$-st-special, and some Aronszajn tree is not $S$-st-special whenever $S - S^*$ is stationary.
Consequently, there is a model of ZFC+GCH+SH+ not every Aronszajn tree is special