The special tree number

Abstract

Define the special tree number, denoted st, to be the least size of a tree of height ω1 which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of MA, but in this paper we look at its relation to other, more typical, cardinal characteristics. Classical facts imply that ℵ1 ≤ st ≤ 2 ℵ 0 , under Martin's Axiom st = 2 ℵ 0 , and that st = ℵ1 is consistent with MA(Knaster)+2 ℵ 0 = κ for any regular κ, thus the value of st is not decided by ZFC and in fact can be strictly below essentially all well studied cardinal characteristics. We show that conversely it is consistent that st = 2 ℵ 0 = κ for any κ of uncountable cofinality, while non(M) = a = s = g = ℵ1.In particular, st is independent of the left hand side of Cichoń's diagram, amongst other things. The proof involves an in-depth study of the standard ccc forcing notion to specialize (wide) Aronszajn trees, which may be of independent interest.