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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There is a natural way to formulate fragments of Todorcevic’s strong reflection principle (SRP) which are associated to forcing classes more restrictive than the class of all stationary set preserving forcing notions. The fragment associated to the subcomplete forcings (SC-SRP), while retaining many crucial consequences of SRP, is compatible with CH, and even Jensen's Diamond Principle. In particular, the saturation of the nonstationary ideal, a celebrated consequence of SRP, does not follow from its subcomplete fragment. In fact, adding CH to SC-SRP results in a principle which outright contradicts the saturation of the nonstationary ideal. A specific form of diagonal reflection of stationary sets of ordinal was used by Paul Larson to separate SRP from Martin's Maximum: that form of diagonal reflection follows from MM, but not from SRP. The surprising initial observation is that it does follow from SC-SRP + CH. The key reason for this is that SC-SRP + CH implies the nonsaturation of the nonstationary ideal. Thus, an apparent weakness of SC-SRP + CH turns out to be a strength in this context.

I will introduce the concepts involved and present some further results along these lines. The picture that emerges is that in the context of SC-SRP, saturation and diagonal reflection work against each other.

This is joint work with Hiroshi Sakai.

Tennenbaum's theorem states that no non-standard model of PA is computable. Hence, no unsound extension of PA has computable models. Pakhomov recently showed that this consequence of Tennenbaum's theorem is fragile; it depends on the signature in which PA is presented. In particular, there is a theory T such that (i) T is definitionally equivalent to PA (this is a strong form of bi-interpretability) and (ii) every consistent r.e. extension of T has a computable model. Pakhomov's techniques yield analogous results for ZF and other canonical systems. He asked whether there is a consistent, r.e. theory T such that no theory which is definitionally equivalent to T has a computable model. We answer this question with an ad hoc construction. This is joint work with Patrick Lutz.

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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Fundamental to the practice of logic is the dogma regarding the first order/second order logic distinction, namely that it is ironclad. Was it always so? The emergence of the set theoretic paradigm is an interesting test case. Early workers in foundations generally used higher order systems in the form of type theory; but then higher order systems were gradually abandoned in favour of first order set theory—a transition that was completed, more or less, by the 1930s. In this talk I will look at first order logic from various points of view, arguing that the distinction between first order and higher order logics, such as second order logic, is somewhat context dependent. From the philosophical or foundational point of view this complicates the picture of first order logic as a canonical logic.

A dimension group is a partially ordered Abelian group whose partial order is isolated and directed and has the Riesz interpolation property. A dimension group is simple just in case it has no nontrivial ideals, ideals being directed convex subgroups. By concentrating on the behavior of positive formulas in simple dimension groups, this talk will reveal a well-behaved part of their model theory.

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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I want to argue that when knowing the model-theory of categories, you kind of know the model-theory of any structure. As the ? at the end of the title suggests, some of this is still speculative.

It is easy to see a category as a first-order structure in the two-sorted language (for objects and morphisms) of categories; a little less to do this foundationally correct (I have given a talk a way back in which I ignored these issues, but I will correct this in the talk, although not mentioning them in this abstract). Now, to any theory T in some first-order language L, we can associate a theory in the language of categories, cat(T), which reflects this theory: the models of cat(T) are isomorphic (as categories) with subcategories of the category Mod(T) of models of T. In fact, any category that is elementary equivalent with Mod(T) is a sub-model of the latter.

This translation from T into cat(T)---from an arbitrary signature to a fixed one---is still mysterious, and as of now, I only know a very few concrete cases. A key role seems to be played by the theory FO, consisting of all sentences in the language of categories which hold in each category of L-structures, for all possible languages L. But I do not even know yet a full axiomatization of FO.

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