NYLogic

The talk will begin by discussing the basic definitions and general goals behind the theory of Borel equivalence relations. We focus on the Friedman-Stanley jumps $=^{+n}$, for $n=1,2,...$ and $n=\omega$. These Borel equivalence relations represent the notions of being classifiable using invariants which are countable sets of reals, countable sets of countable sets of reals, and so on. We consider the problem of constructing a Borel reduction from $=^{+n}$ to some other equivalence relation.

For $n=1$ the situation is well understood and there are many such results. For example: Marker proved that for a first order theory with an uncountable type space, its isomorphism relation is above $=^{+1}$; Larson and Zapletal characterized the analytic equivalence relations above $=^{+1}$ as those which are 'unpinned' in the Solovay extension.

In this talk we present a new technique for proving that an equivalence relation is above $=^{+n}$, when $n>1$, based on Baire-category methods. As corollaries, we conclude that $=^{+\omega}$ is 'regular' (answering a question of Clemens), and that $=^{+n}$ is 'in the spectrum of the meager ideal' (extending a result of Kanovei, Sabok, and Zapletal for $n=1$).

This first meeting will be partially devoted to organizing for the semester. But, I will also begin talking about Jamshid Derakhshan' survey paper on the model theory of the adeles entitled 'Model Theory of the Adeles and Number Theory'.

This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.

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This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.

Slides

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This talk, in the Mathematics Department Colloquium of the CUNY Graduate Center, will be aimed at a broad mathematical audience.
Traditionally, computability theory has been restricted to countable structures (such as groups or rings). We explain how digital computation by Turing machines can be applied to continuum-sized structures, with particular attention to the real numbers and the absolute Galois group of the rationals, and present some natural and intriguing questions regarding each.

We will discuss the paper of Cluckers, Derakhshan, Leeknegt and Macintyre on uniformly defining valuation rings in Henselian valued fields with finite or pseudofinite residue fields.

Expansions of the ordered additive group of the reals (or more generally definably complete expansions of ordered Abelian groups) of finite dp-rank are a class of reasonably well-behaved ordered structures that generalize the class of o-minimal structures. In this talk I will give a survey of ongoing work with John Goodrick on exploring the properties of definable sets in this class of structures.

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We discuss the famous open problem of determining the exact consistency strength of PFA. We present an equiconsistency between Ben Goodman's Sigma_n-Correct Proper Forcing Axiom, which implies PFA, and supercompact for C^(n-1)-cardinals under additional mild assumptions for large enough n. Without these assumptions we can prove a dichotomy resembling Woodin's HOD dichotomy with a model containing the mantle taking on the role of HOD.

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In recent work with Luca Motto Ros we prove that under analytic determinacy there exists an analytic relation that is not class-wise Borel embeddable into any orbit equivalence relation. The result builds on an unpublished result of Becker from 2001 and fits in the area of invariant descriptive set theory. I will mainly discuss our result and how it is related to a major conjecture in the field known as the '$E_1$ conjecture'.

I will continue talking about Derakhsan's survey article 'Model Theory of Adeles and Number Theory'.

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Polish spaces, the objects of study of descriptive set theory, are completely metrizable topological spaces that have a countable dense subset. For example, the reals - the first Polish space in the world. We will look at 'sigma-ideals' of compact subsets of a Polish space. Think of a sigma-ideal as being a collection of 'small' compact sets, under some notion of smallness -- so for example, your Polish space could be the interval $[0,1]$ and your sigma-ideal could be the collection of all its compact sets of Lebesgue measure $0$. The descriptive-set-theoretic study of these objects yields rich results for the following reason. If you look at the collection of all the compact subsets of a Polish space, that too, topologized and metrized in a natural way, turns out to be a Polish space. This means that you can look at your sigma-ideal of compact sets in two places: in the original space, say $E$, and in the `hyperspace' of all compact sets of $E$. In this talk we will deal with sigma-ideals that can be represented in a very nice way inside this hyperspace, and we will examine the behaviour of so-called G-delta subsets of $E$ with respect to this representation.

We prove that a random group, in Gromov's density model with $d\lt 1/2$, satisfies an AE sentence (in the language of groups) if and only if this sentence is true in a nonabelian free group. This is a joint work with R. Sklinos.

We present new results regarding the depth and Tukey spectrum of general ultrafilters and simple
$P_\lambda$-points at a measurable cardinal. In particular we prove that on a measurable cardinal there can only be a single $\lambda$ for which there exists a simple $P_\lambda$-point - this is in sharp contrast to $\omega$. Finally we will present several models in which we analyze the depth and Tukey spectrum of an ultrafilter, and their effect on generalized cardinal characteristics.

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A few years ago, a problem arose in some of my work that I wasn’t able to solve, forcing me to add a technical hypothesis to a theorem - this has bothered me ever since. The issue has to do with the relationship between independence in a stable (or simple or NSOP_1) theory and independence in a stable reduct. In this rather informal talk, I will describe the problem and some partial results. The audience is welcome to provide proofs or counterexamples.

Recall that a structure is pseudofinite if every sentence satisfied by that structure has a finite model - equivalently, if the structure is elementarily equivalent to an ultraproduct of finite structures. In this talk, I will present some work in progress from two independent projects around pseudofinite rings and pseudofinite modules: one is joint work with Alex Van Abel, the other is work of my PhD student Roberto Torres. These two projects are linked by the important role played by the class of von Neumann regular rings.

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The relationship between left distributivity and very large cardinal embeddings was discovered in the 1980s but remains, in many ways, mysterious. In the late 1980s Richard Laver showed that the closure of a single elementary embedding under the application operation generates a free left distributive algebra and demonstrated the linearity of a particular ordering on terms of the free left distributive algebra (given the existence of such embeddings). Patrick Dehornoy later used the braid group on infinitely many generators to show the linearity of that ordering relation within ZFC. (The consistency strength of other related theorems is still unknown). David Larue subsequently extended that work to demonstrate braid group representations of the free left distributive algebra on $n$ generators, for any natural number $n$. Still elusive was an algebra of embeddings isomorphic to a free left distributive algebra on more than one generator. We present an inverse limit construction of such a free, two-generated left distributive algebra of embeddings from a slightly stronger large cardinal assumption than the one used by Laver. (Joint work with Andrew Brooke-Taylor and Scott Cramer.)

We will consider the space of functions from $\lambda$ to $\kappa$ for various choices of $\lambda$ and $\kappa$. In the first part of the talk we define topologies on such spaces and discuss the $\mu$-meagre ideal (i.e. sets that are unions of $\mu$-many nowhere dense sets), and their associated cardinal invariants. In the second part, we will look at various ways to consider (cardinal invariants of) the null ideal on such spaces.

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In this talk, I explore the connections between Statistical Learning Theory and Model Theory. This includes the connections between PAC-learning and NIP and the connections between differentially private PAC-learning and stability. Finally, I examine the work that my colleagues and I have started on improving the sample complexity of differentially private PAC-learning algorithms using techniques from stability theory. This work is joint with Alexei Kolesnikov, Miriam Parnes, and Natalie Piltoyan.

In this talk we discuss automorphisms of countable short recursively saturated models of PA.

Kossak-Schmerl 95 shows that: if M is a countable, arithmetically saturated model of PA, then the automorphism group of M codes its standard system. We discuss how to prove a similar result for countable short arithmetically saturated models of PA.

This is joint work with Erez Shochat.

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We identify several examples of distal metric structures and examine several consequences of distality, such as the existence of distal cell decompositions, in each. These results include joint work with Itaï Ben Yaacov and with Diego Bejarano. One class of examples starts with finding a metric structure whose automorphism group is the group of increasing homeomorphisms of the unit interval. We will discuss some properties of this structure and extrapolate to other models of its theory, which we call 'dual linear continua.' Another source of examples includes real closed metric valued fields. These give rise to a notion of ordered metric structure, providing a viewpoint to study o-minimality in continuous logic.

We show that, given a reflecting cardinal, one can generically produce a universe of $\mathsf{BPFA}$ in which additionally the $\Sigma^1_{n+2}$-uniformization property holds for every $n$ simultaneously.

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In this talk we continue the discussion on the automorphism groups of countable short recursively saturated models of PA. In particular, we discuss the cuts of the model which are fixed setwise by all automorphisms (invariant cuts). We show that such cuts occur in different places of the model, depending on the types realized in the last gap. We then show that this implies, in some of these cases, that the automorphism groups of such models are non-isomorphic as topological groups. This is a joint work with Ermek Nurkhaidarov.

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We consider a computability-theoretic version of the ultraproduct construction for an infinite uniformly computable sequence of structures, where the role of an ultrafilter is played by an infinite set of natural numbers that cannot be split into two infinite subsets by any computably enumerable set. For computable structures, effective powers preserve only the first-order sentences of lower levels of quantifier complexity. Additional decidability of the structure increases preservation of the fragments of its theory in an effective power, so that a structure with a computable elementary diagram is elementarily equivalent to its effective power. We will present a number of recent collaborative results.

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I shall expound Michael Dummett's claim in his paper, 'The philosophical significance of Gödel's theorem' (1963), and in later writings, that a consequence of the indefinite extensibility of Gödel incompleteness is that 'the meaning of 'natural number' is inherently vague'. Though of course Gödel incompleteness establishes that every formal system containing basic arithmetic has a proper extension, I claim, against Dummett's view, that there is a notion of arithmetical truth intrinsic to the meaning of 'natural number' which is stable, not indefinitely extensible, and that first-order Peano Arithmetic is sound and complete with respect to that notion of arithmetical truth. Thereby the meaning of 'natural number' is not vague but clear and precise.

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Given a topological space $(X, \tau)$, the derived set operator $d_\tau$ maps a set $T$ to its set of limit points with respect to $\tau$. Fixing an initial topology $\tau_0$ on $X$, we can define a sequence of derived topologies $(X, \tau_0, \tau_1, \dots, \tau_\xi, \dots)$, where $\tau_\zeta \subseteq \tau_\xi$ for $\zeta \lt \xi$. This is achieved by declaring $d_{\tau_\xi}(T)$ to be open in $\tau_{\xi+1}$ and taking unions at limit stages.

In Derived Topologies on Ordinals and Stationary Reflection, Bagaria characterised the non-isolated points in the $\xi$-th derived topology on ordinals as those satisfying a strong iterated form of stationary reflection, termed $\xi$-simultaneous reflection.

Generalisations of combinatorial properties of ordinals to $\mathcal{P}_\kappa(A) := \{X \subseteq \kappa : |X| \lt \kappa\}$, where $\kappa$ is an uncountable regular cardinal and $A \subseteq \kappa$, have been widely studied. In this context, we extend the notion of higher stationarity and construct a sequence of topologies $\langle \tau_0, \tau_1, \dots \rangle$ on $\mathcal{P}_\kappa(A)$, characterising the simultaneous reflection of high-stationary subsets of $\mathcal{P}_\kappa(A)$ in terms of elements in the base of a derived topology on $\mathcal{P}_\kappa(A)$.

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