Set Theory Seminar

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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A central question in the theory of large cardinals was whether the existence of a model of ZFC with exactly two normal measures follows from the consistency of ZFC with a measurable cardinal. This was answered positively by a landmark theorem of Friedman and Magidor, whose proof masterfully combined advanced techniques in the theory of large cardinals, including generalized Sacks forcing, forcing over canonical inner models, coding posets, and nonstationary support iterations.

In this talk, we present a new and simpler proof of the Friedman-Magidor theorem. A notable feature of our approach is that it avoids any use of inner model theory, making it applicable in the presence of very large cardinals that are beyond the current reach of the inner model program. If time permits, we will also discuss additional applications of the technique: the construction of ZFC models with several normal measures but a single normal ultrapower; a nontrivial model of the weak Ultrapower Axiom from the optimal large cardinal assumption; and a generalization of the Friedman–Magidor theorem to extenders.

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We put together Woodin's $ \Sigma^2_1$ basis theorem of $AD^+$ and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(\Sigma^2_1)^{uB}$ statement that is true in V is true in HOD. Moreover, this is true even if we allow a certain parameter. We then show that stronger absoluteness cannot be implied by any large cardinal axiom consistent with the axiom V = Ultimate L.

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