NYLogic

I will talk about weak embeddability and the universality number of the class of Aronszajn trees, with a focus on the role of specializing triples.

The notion of a specializing triple was introduced by Džamonja and Shelah in their strong negative solution to an old problem on the existence of a universal (with respect to weak embeddability) wide Aronszajn tree under Martin's axiom. Their proof has two stages: first, they reprove a theorem of Todorčević showing that under ${\rm MA}_{\omega_1}$ there is no universal Aronszajn tree, and then they show that every wide Aronszajn tree weakly embeds into an Aronszajn tree. The second stage involves a rather complicated ccc forcing. However, already in the first stage, they introduce a new technique: the notion of a specializing triple, and prove that for each Aronszajn tree $T$, there is a ccc forcing adding another Aronszajn tree $T^*$ together with a specializing function on $T^*\otimes T$ such that $(T^*, T, c)$ is a specializing triple. In particular, this shows that $T^*$ does not weakly embed into $T$.

I will explain how a slight but careful modification of this definition makes it possible to accommodate wide trees directly, yielding a more streamlined proof of Džamonja and Shelah’s result. More precisely, for every $\kappa$-wide Aronszajn tree $T$, there is a ccc forcing adding an Aronszajn tree $T^*$ and a function $c$ such that $(T^*, T, c)$ is what I call a left specializing triple. From this, one quickly recovers Džamonja-Shelah’s theorem: under Martin’s axiom, every class of trees of height $\omega_1$ and size less than the continuum but with no cofinal branches either is not universal for Aronszajn trees, or has universality number equal to the continuum.

Finally, I will indicate how the modified definition can also be used to show that this consequence of Martin’s axiom is consistent with the existence of a nonspecial Aronszajn tree.

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