Forcing Axioms and the Continuum Hypothesis, part II: Transcending ω₁-sequences of real numbers

The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah. The corresponding completely proper forcing which can be constructed using CH is moreover a tree whose square is special off the diagonal. While such trees had previously been constructed by Jensen and Kunen under the assumption of Jensen's diamond principle, this is the first time such a construction has been carried out using the Continuum Hypothesis.