Abstract. We propose a parameterized proxy principle from which κ-Souslin trees with various additional features can be constructed, regardless of the identity of κ. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent κ-Souslin tree that applies also for κ inaccessible.We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known ♦-based constructions of κ-Souslin trees may be redirected through this new proxy principle.

We study the relationship between a κ-Souslin tree T and its reduced powers T θ / . Previous works addressed this problem from the viewpoint of a single power θ , whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an ℵ 6 -Souslin tree T and a sequence of uniform ultrafilters n | n < 6 such that T ℵn / n is ℵ 6 -Aronszajn if and only if n < 6 is not a prime number. This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

No abstract

No abstract