# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Edmund Husserl’s Philosophy of Arithmetic

Edmund Husserl, the principal founder of phenomenology, is well known for his contributions to philosophy. What is less known is his early work in mathematics. Husserl studied under Kronceker and Weierstrass in Berlin, and got his Ph.D. in 1883 in Vienna working on the calculus of variations under supervision of Leo Königsberger. Under influence of Franz Brentano, he moved to philosophy, and in from 1901 to 1916 he taught philosophy in Göttingen, where he interacted with Hilbert. In “Logic and Philosophy of Mathematics in the Early Husserl” Springer 2010, Stefania Centrone analyses Husserl’s work on foundations of mathematics from that period, and shows its connections to Hilbert’s ideas. I will say a few words about phenomenology, and I’ll talk about fragments of Husserl’s “Philosophy of Arithmetic.”

# No Logic Workshop March 4.

We do not have a speaker for Friday, March 4, so there will be no Logic Workshop that day. (The Set Theory and Model Theory Seminars will meet as usual.)

# Trichotomy principle for partial differential fields

The Zilber trichotomy principle gives a precise sense in which the structure on a sufficiently well behaved one-dimensional set must have one of only three possible kinds: disintegrated (meaning that there may be some isolated correspondences, but nothing else), linear (basically coming from an abelian group with no extra structure), or algebro-geometric (essentially coming from an algebraically closed field). This principle is true in differentially closed fields when “one dimensional” is understood as “strongly minimal” (proven by Hrushovski and Sokolovic using the theory of Zariski geometries and then by Pillay and Ziegler using jet spaces).

When working with differentially closed fields with finitely many, but more than one, distinguished commuting derivations, there are sets which from a certain model theoretic point of view (having to do with the notion of a regular type) are one dimensional even though they are infinite dimensional from the point of view of differential dimension. Moosa, Pillay and Scanlon showed that a weakening of the trichotomy principle is true for these sets: if there is a counter example to the trichotomy principle, then one can be found for a set defined by linear PDEs.

In this lecture, I will explain in detail what the trichotomy principle means in differential algebra, how the reduction to the linear case works, and then how one might approach the open problems.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.

# Classification of strongly normal extensions of a differential field, and related issues

The material is taken from a joint paper with M. Kamensky, “Interpretations and differential Galois extensions.” Given a differential field *K* with field of constants *k*, and a logarithmic differential equation over *K*, the strongly normal extensions of *K* for the equation correspond (up to isomorphism over *K*) with the connected components of *G(k)* where *G* is the Galois groupoid of the equation. This generalizes to other contexts (parameterized theory,….), and is also the main tool in existence theorems for strongly normal extensions with prescribed properties.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.

# Studying profinite monoids via logic

This talk is about my ongoing joint research project with Benjamin Steinberg (CCNY). We begin with the observation that the free profinite aperiodic monoid over a finite set *A* is isomorphic to the Stone dual space (spectrum) of the Boolean algebra of first-order definable sets of finite *A*-labelled linear orders (“*A*-words”). This means that elements of this monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite *A*-words. We exploit this view of the free profinite aperiodic monoid to prove both old and new things about it using methods from model theory, in particular (weakly) saturated models.

The talk is aimed at anyone with a basic knowledge of model theory, not necessarily of profinite monoids; in particular I will take care to review some background on profinite monoids and on how they relate to logic and regular languages.

# What we talk about when we talk about truth

The goal of set theory, as articulated by Hugh Woodin in his recent Rothschild address at the Isaac Newton Institute, is develop a “convincing philosophy of truth.” There he described the work of set theorists as falling into one of two categories: studying the universe of sets and studying models of set theory. We offer a new perspective on the nature of truth in set theory that may to some extent reconcile these two efforts into one. Joint work with Shoshana Friedman.

# The definability of radicals in noncommutative rings

We will survey some results in the definability of radicals in rings, focusing on some recent results about noncommutative rings. In particular, we will show that there is no simple definition of the prime radical in the noncommutative setting, thus differentiating prime radicals in commutative/noncommutative rings.

# The model theory of some j-functions

The model theory of j-functions (and more generally of modular and Shimura curves) has been studied by Adam Harris and Christopher Daw. They connected in an intriguing way the categoricity of (an infinitary theory of) the j-function with results in arithmetic geometry (a version of the Mumford-Tate conjecture). I will discuss some of these connections and the questions they raise for model theory, especially in connection with the quest for new versions of j-functions (e.g. on real fields).

# Woodin’s AD-conjecture and local Reinhardt cardinals

We will discuss Woodin’s AD-conjecture, which gives a deep relationship between very large cardinals and determined sets of reals. In particular we will show that the AD-conjecture holds for the axiom I0 and that there are many interesting consequences of this fact. We will also discuss the notion of a local Reinhardt cardinal and how variations of the AD-conjecture might show that such cardinals do not exist.

# Boolean ultrapowers in set theory

Boolean ultrapowers can serve to explain phenomena that arise in the context of iterated ultrapowers, such as the genericity of the critical sequence over the direct limit model. The examples we give are ultrapowers formed using the complete Boolean algebras of Prikry forcing, Magidor forcing, or a generalization of Prikry forcing. Boolean ultrapowers can also be viewed as a direct limit of ultrapowers, and we present some criteria for when the intersection of these ultrapowers is equal to the generic extension of the Boolean ultrapower, thus arriving at a generalization of a phenomenon first observed by Bukovsky and Dehornoy in the context of Prikry forcing.

# Hilbert’s Tenth Problem inside the rationals

For a ring *R*, Hilbert’s Tenth Problem is the set *HTP(R)* of polynomials *f ∈ R[X _{1},X_{2},…]* for which

*f=0*has a solution in

*R*. Matiyasevich, completing work of Davis, Putnam, and Robinson, showed that

*HTP(*is Turing-equivalent to the Halting Problem. The Turing degree of

**Z**)*HTP(*remains unknown. Here we consider the problem for subrings of

**Q**)*. One places a natural topology on the space of such subrings, which is homeomorphic to Cantor space. This allows consideration of measure theory and also Baire category theory. We prove, among other things, that*

**Q***HTP(*computes the Halting Problem if and only if

**Q**)*HTP(R)*computes it for a nonmeager set of subrings

*R*.

# Loftiness

Loftiness is a weak notion of saturation. It was defined and studied in detail by Matt Kaufmann and Jim Schmerl in two substantial papers published in 1984 and 1987. Kaufman and Schmerl discovered that there are many shades of loftiness. I will give an overview of model theory of lofty models of arithmetic and I will talk about constructions of lofty models that are not recursively saturated.

# Intrinsic density and computability

The idea of a negligible subset of the natural numbers is highly appealing; bad behavior often seems restricted to a “small” set. Asymptotic density (the usual solution) has been used repeatedly in many fields, most prominently in number theory. Unfortunately, this is largely incompatible with the computability theorist’s perspective; even 1-equivalent sets can have wildly different densities. However, by restricting to cases where density is computably invariant, we develop a new notion with appealing properties. We will characterize exactly how hard it is to make a set with intrinsic density, and uncover connections to both computability and randomness. In particular, we suggest that Post missed a very natural immunity property when developing his notions of hyperimmunity.

# Categories turned models: taming the finite

Whereas a category theorist sees mathematics as objects interacting with each other via maps, a model theorist looks instead at their internal structure. So we may think of the former as the sociologues of mathematics and the latter as their psychologues. It is well-known that to a first-order theory we can associate the category of its models, but this produces often a non-natural category, as the maps need to be elementary, and maps rarely are! I will discuss the opposite (Jungian?) perspective: viewing a category as a first-order structure. This yields some unexpected rewards: it allows us to define certain second-order concepts, like finiteness, in a first-order way. I will illustrate this with some examples: sets, modules, topologies, …

# In search of a complete axiomatization for $F_p((t))$

I will give a survey of the attempts that have been made since the mid 1960’s to find a complete recursive axiomatization of the elementary theory of $F_p((t))$. This problem is still open, and I will describe the difficulties researchers have met in their search. Some new hope has been generated by Yu. Ershov’s observation that $F_p((t))$ is an “extremal” valued field. However, while his intuition was good, his definition of this notion was flawed. It has been corrected in a paper by Azgin, Kuhlmann and Pop, in which also a partial characterization of extremal fields was given. Further progress has been made in a recent manuscript, on which I will report at the AMS meeting at Rutgers. The talk at the Graduate Center will provide a detailed background from the model theoretic point of view.

The property of being an “extremal valued field” is both elementary and very natural, so it is an ideal candidate for inclusion in a (hopefully) complete recursive axiomatization for $F_p((t))$. It implies an axiom scheme that was considered previously, which describes the behavior of additive polynomials under the valuation. I will discuss why additive polynomials are crucial for the model theory of valued fields of positive characteristic.

The open problems around extremal fields provide a good source of research projects of various levels of difficulty for young researchers.

This talk is jointly sponsored by the Commutative Algebra & Algebraic Geometry Seminar and the CUNY Logic Workshop.

# Open determinacy for games on the ordinals is stronger than ZFC

The principle of open determinacy for class games — two-player games of perfect information with plays of length ω, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates TR_α for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC + Pi^1_1 comprehension, a proper fragment of Kelley-Morse set theory KM.

This is joint work with Victoria Gitman, with helpful participation of Thomas Johnstone.

See also related article: V. Gitman, J.D. Hamkins, Open determinacy for class games, submitted.

For further information and commentary concerning this talk, please see the related post on my blog.

# Theories where any definable infinite set has interior

We consider the case of a theory $T$ which expands that of divisible ordered Abelian groups and has the property that in any model of $T$ any infinite definable subset has non-empty interior (in the order topology). We will call such theories visceral. Visceral theories arise naturally in the study theories with a strong form of the independence property and generalize the class of o-minimal and weakly o-minimal theories. We consider the structure of definable sets and definable functions in visceral theories, giving some weak structural results. We also consider how to build general classes of examples of visceral theories, and relate these example back to questions about strong forms of the independence property.

# On computing VC-density in VC-minimal theories

In model theory, theories are typically distinguished by the complexity of their definable families. One popular notion of complexity, Vapnik-Chervonenkis density, is borrowed from statistical learning theory. In this talk, I discuss the general notion of computing VC-density in NIP theories, a notion explored by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko in recent work. In this work, they ask if there is a relationship between dp-rank and VC-density. I show a partial result pointing in that direction by studying VC-minimality (a condition stronger than having minimal dp-rank). Any formula in a VC-minimal theory with two parameter variables has VC-density at most two. I conclude by discussing the possibility of extending this result to higher dimensions.

# Model theory of right-angled buildings

# No talks November 27

There will be no talks on November 27, the day after Thanksgiving.