A cofinality-preserving small forcing may introduce a special Aronszajn tree

Extended Abstract:

• Shelah proved that Cohen forcing introduces a Souslin tree;
• Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree;
• Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis;
• Irrgang introduced a c.c.c. notion of forcing based on a simplified (${\omega }_1$,1)-morass that adds an ${\omega }_2$-Souslin tree.

Here, it is proved that adding a subset of ${\omega }_2$ may introduce a special Aronszajn tree of height ${\mathrm{\aleph }}_{{\omega }_1+1}$ :

Starting with a model of two supercompact cardinals, we construct a model with no special ${\mathrm{\aleph }}_{{\omega }_1+1}$-Aronszajn trees, in which there exists a notion of forcing $\mathbb{P}$ of cardinality ${\omega }_3$, which is $\sigma$-closed, ${\omega }_1$-distributive, ${\omega }_3$-Knaster, and such that in the generic extension by $\mathbb{P}$, there exists a special Aronszajn tree of height ${\mathrm{\aleph }}_{{\omega }_1+1}$.

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Citation information:

A. Rinot, A cofinality-preserving small forcing may introduce a special Aronszajn tree, Arch. Math. Logic, 48(8): 817-823, 2009.