Chang's Conjecture, The Weak Reflection Principle and the Tree Property at ${\omega }_2$

Abstract

We prove that a strong version of Chang's Conjecture, equivalent to the Weak Reflection Principle at ${\omega }_2$, together with $2^{\omega }={\omega }_2$, imply there are no ${\omega }_2$-Aronszajn trees.

We prove indeed that ${\mathrm{CC}}^{\ast }+\neg \mathrm{CH}$ imply the Tree Property for ${\omega }_2$ holds ($\mathrm{TP}({\omega }_2)$). However, there was an error in our claim. It is not true that ${\mathrm{CC}}^{\ast }$ is equivalent to the Weak Reflection Principle at ${\omega }_2$ ($\mathrm{WRP}({\omega }_2)$). The question if $\mathrm{WRP}({\omega }_2)+\neg \mathrm{CH}\to \mathrm{TP}({\omega }_2)$ remains open

References in corpus (1)

• Stationary reflection principles and two cardinal tree properties

Cited by in corpus (3)

• Separating Subversion Forcing Axioms
• Adjoining only the things you want: a survey of Strong Chang's Conjecture and related topics
• A variant of Shelah's characterization of Strong Chang's Conjecture