SOME RESULTS IN THE EXTENSION WITH A COHERENT SUSLIN TREE (Aspects of Descriptive Set Theory)

We show that under PFA $(S)$ , the coherent Suslin tree $S$ (which is a witness of the axiom PFA $(S)$ ) forces that there are no $\omega_{2}$ -Aronszajn trees.We also determine the values of cardinal invariants of the continuum in this extension. INTRODUCTIONIn [20], Stevo Todor\v{c}evi\v{C} introduced the forcing axiom PFA $(S)$ , which says that there exists a coherent Suslin tree $S$ such that the forcing axiom holds for every proper forcing which preserves $S$ to be Suslin, that is, for every proper forcing $\mathbb{P}$ which preserves $S$ to be Suslin and $\aleph_{1^{-}}$ many dense subsets $D_{\alpha},$ $\alpha\in\omega_{1}$ , of $\mathbb{P}$ , there exists a filter on $\mathbb{P}$ which intersects all the $D_{\alpha}$ .PFA $(S)[S]$ denotes the forcing extension with the coherent Suslin tree $S$ which is a witness of PFA $(S)$ .Since the preser- vation of a Suslin tree by the proper forcing is closed under countable support iteration (due to Tadatoshi Miyamoto [15]), it is consistent relative to some large cardinal assumption that PFA $(S)$ holds.The first appearance of such a forcing axiom is in the paper [13] due to Paul B. Larson and Todor\v{c}evi\v{c}.In this paper, they introduced the weak version of PFA $(S)$ , called Souslin's Axiom (in which the properness is replaced by the cccness), and under this axiom, the coherent Suslin tree $S$ , which is a witness of the axiom, forces a weak fragment of Martin's Axiom.In [20], it is also proved that under PFA $(S),$ $S$ forces the open graph dichotomy () and the P-ideal dichotomy.Namely, many consequences of PFA are satisfied in the extension with $S$ under 2000 Mathematics Subject Classification.$03E50,03E05,03E35$ .

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