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.

The Partition Principle (PP) states that if there is a surjection A to B then there is an injection B to A. While this is an immediate consequence of the Axiom of Choice (AC), the question of if PP implies AC is one of the longest-standing open questions in set theory. Partial results regarding this come to us from many sources, including a theorem of Pincus that tells us that if 'for all ordinals A and all sets B, if there is a surjection B to A then there is an injection A to B' implies AC for well-orderable families of sets. We shall dissect this and related results, looking into the history of the structure of the cardinals in choiceless models and following the throughline to modern research on eccentric sets and the structure of cardinals as a partial order.

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Gitman, Friedman, and Kanovei described a construction of a model of ZF in which countable choice holds and $\Pi^1_2$-dependent choice for reals is not satisfied. We modify that construction to obtain a model in which (boldface) $\Pi^1_n$-determinacy holds, but which fails to satisfy (lightface) $\Pi^1_{n+2}$-${\rm DC}$ for reals. In particular, we show that no projective level of determinacy implies full ${\rm DC}_{\mathbb R}$. This is joint work with Sandra Müller.

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In this talk we will discuss some problems regarding adding linear order isomorphisms between dense sets of reals via forcing. A set of reals $A \subseteq \mathbb R$ is called $\aleph_1$-dense just in case it has intersection size $\aleph_1$ with every nonempty open interval. Famously Baumgartner showed in the 70s that consistently all $\aleph_1$-dense sets of reals are order isomorphic, thus establishing the consistency of the natural analogue of Cantor's categoricity theorem for countable dense linear orders at the uncountable. He later showed the same statement follows from PFA. The statement 'all $\aleph_1$-dense sets of reals are isomorphic' is now known as Baumgartner's axiom and denoted BA.

Baumgartner's argument is famously tricky and has several interesting features. Given $\aleph_1$-dense sets $A$ and $B$ he shows that under the continuum hypothesis there is always a ccc partial order for making them isomorphic - but here the CH is important (though it must fail in the final model). Obviously it is therefore natural to ask whether the CH is necessary and, similarly whether BA follows already from MA and not just PFA. Avraham and Shelah showed the answer is 'no' to both: they produced a model of MA in which there is an $\aleph_1$-dense set of reals $A$ so that no ccc forcing notion can add an isomorphism between $A$ and its reverse ordering. In the first part of this talk we will strengthen this result by showing that MA is in fact consistent with an $\aleph_1$-dense linear order $A$ so that any partial order of size $\aleph_1$ which adds an isomorphism between $A$ and its reverse ordering must collapse $\aleph_1$. Thus it is consistent with MA that no such order can even be proper. This part is joint work with Pedro Marun and Saharon Shelah.

In another direction there is a natural generalization of BA to non-ordered spaces, most notably higher dimensional Euclidean spaces $\mathbb R^n$ for $n > 1$ as well as compact $n$-dimensional manifolds. Curiously, in these cases Steprans and Watson showed that the corresponding BA statements do indeed follow from MA so the case of dimension one is unique. In particular BA for $\mathbb R^n$ with $n > 1$ does not imply the one dimensional case. They then conjectured that conversely BA implies its higher dimensional analogues. In the second part of the talk we will introduce some intrigue to this conjecture by showing any 'reasonable' way of forcing BA - a very general adjective that includes all known methods - must necessarily force the higher dimensional versions and much more including large fragments of MA.

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A Borel graph is hyperfinite if it can be written as a countable increasing union of Borel graphs with finite components. It is a major open problem in descriptive set theory to determine the complexity of the set of hyperfinite Borel graphs. In a recent paper, Conley, Jackson, Marks, Seward, and Tucker-Drob introduce the notion of Borel asymptotic dimension, a definable version of Gromov's classical notion of asymptotic dimension, which strengthens hyperfiniteness and implies several nice Borel combinatorial properties. We show that the set of locally finite Borel graphs having finite Borel asymptotic dimension is $\Sigma^1_2$-complete. This is joint work with Jan Grebik.

Petr Vopěnka (1935-2015) was an influential mathematician and deep and original thinker. In the 1970s, he and Petr Hájek developed a system of non-Cantorial set theory, wich was first known the theory of semi-sets, and later became Alternative Set Theory (AST). The primary new concept on which the axioms of AST are based is what Vopěnka called natural infinity. In my talk I will outline some of Vopenka's thoughts on foundations of mathematics and the role of set theory in it, followed by a discussion of the axioms of AST.

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