Abstract

Abstract. The tree property at κ + states that there are no Aronszajn trees on κ +, or, equivalently, that every κ + tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ + and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper we reconcile the two. We prove from large cardinals that the tree property at κ + is consistent with failure of the singular cardinal hypothesis at κ. §1. Introduction. In the early 1980s Woodin asked whether failure of the singular cardinal hypothesis (SCH) at ℵω implies the existence of an Aronszajn tree on ℵω+1. More generally, in 1989 Woodin and others asked whether failure of the SCH at a cardinal κ of cofinality ω, implies the existence of an Aronszajn tree on κ +, see Foreman [7, §2]. To understand the motivation for the question let us