We show that, under PFA, a coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic.

A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with a review (...) of some generalizations of results from descriptive set theory concerning Baire category which are then used in non-trivial Souslin tree constructions that yield Souslin algebras with a remarkable subalgebra structure. In particular, we use this method to prove that under the diamond principle there is a bi-embeddable though not isomorphic pair of homogeneous Souslin algebras. (shrink)

We investigate the effect after forcing with a coherent Souslin tree on the gap structure of the class of coherent Aronszajn trees ordered by embeddability. We shall show, assuming the relativized version PFA(S) of the proper forcing axiom, that the Souslin tree S forces that the class of Aronszajn trees ordered by the embeddability relation is universal for linear orders of cardinality at most ${\aleph_1}$.

We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω 1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary (...) ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal. (shrink)

Suppose that T^∗ is an ω_1-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T^∗) for proper forcings which preserve these properties of T^∗. We prove that PFA(T^∗) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ω_1, and the P-ideal dichotomy. On the other hand, PFA(T^∗) implies some of the consequences of diamond principles, such as the existence (...) of Knaster forcings which are not stationarily Knaster. (shrink)

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, or math formula, which asserts that the universe is a c.c.c. forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of math formula. The new axiom is shown to be consistent with the failure of math formula and a singular continuum. We prove that math formula is preserved in a strong way when adding a Cohen (...) real and that adding a random real to a model of math formula preserves math formula. We also consider the analogous variant of the proper forcing axiom. (shrink)

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We (...) also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved. (shrink)