Forcing Axioms and the Continuum Hypothesis, part II: Transcending \omega_1-sequences of real numbers

TL;DR

This paper proves that the forcing axiom for completely proper forcings cannot coexist with the Continuum Hypothesis, resolving a longstanding problem and constructing special trees under CH for the first time.

Contribution

It demonstrates the inconsistency of the forcing axiom for completely proper forcings with CH and constructs special trees using CH, extending previous results.

Findings

Forcing axiom for completely proper forcings is inconsistent with CH.

Constructed a special tree under CH whose square is special off the diagonal.

Resolved a longstanding problem of Shelah regarding forcing axioms and CH.

Abstract

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.