Abstract: It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features are often dismissed as unnatural or as Gödelian trickery. In this talk, I aim to overcome that criticism — as well as I am able to — by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.

The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.

Discussion and commentary can be made on the speaker's web site at http://jdh.hamkins.org/natural-instances-of-nonlinearity-in-the-hierarchy-of-consistency-strength-uwm-logic-seminar-january-2021/.

Abstract: I will speak about joint work with Anand Pillay on the structure of finite approximate groups satisfying a local NIP assumption. Our results can be seen as a unification of the model-theoretic study of "tame" arithmetic regularity with work of Hrushovski and Breuillard, Green, and Tao on the structure theory of approximate groups.

Abstract: The Weihrauch degrees are a popular setting for classifying the computational content of mathematical theorems. Understanding their structure is useful as a technical tool in concrete classifications. Moreover, their structure tells us something about how degrees of non-computability look like in principle. In this talk, I'll summarize what is already known about the structure of the Weihrauch degrees, and try to draw attention to some open problems. For example, we know that they form a distributive lattice, which is not a Heyting algebra and which is not complete. We have further natural algebraic operations, and we know of a few that they are definable in terms of others. The Medvedev degrees embed into the Weihrauch degrees as a lattice, as do the many-one degrees (but in a weird way).

Abstract: According to Felix Klein, “synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates”. To put it less eloquently, the synthetic method axiomatizes geometry directly by construing points and lines as primitive notions, whereas the analytic method builds a model, the Euclidean plane, from the real numbers.

Do other branches of mathematics posses the synthetic method, too? For instance, what would “synthetic topology” look like? To build spaces out of sets, as topologists usually do, is the analytic way. The synthetic approach must construe spaces as primitive and axiomatize them directly, without any recourse to sets. It cannot introduce continuity as a desirable property of functions, for that would be like identifying straight lines as the non-bending curves.

It is indeed possible to build the synthetic worlds of topology, smooth analysis, measure theory, and computability. In each of them, the basic structure – topological, smooth, measurable, computable – is implicit by virtue of permeating everything, even logic itself. The synthetic worlds demand an economy of thought that the unaccustomed mind finds frustrating at first, but eventually rewards it with new elegance and conceptual clarity. The synthetic method is still fruitfully related to the analytic method by interpretation of the former in models provided by the latter.

We demonstrate the approach by taking a closer look at synthetic computability, whose central axiom states that there are countably many countable subsets of the natural numbers. The axiom is validated and explained by its interpretation in the effective topos, where it corresponds to the familiar fact that the computably enumerable sets may be computably enumerated. Classic theorems of computability may be proved in a straightforward manner, without reference to any notion of computation. We end by showing that in synthetic computability, Turing reducibility is expressed in terms of sequential continuity of maps between directed-complete partial orders.

Abstract: In this talk I will show that no infinite field is interpretable in the first-order theory of nonabelian free groups.

Abstract: The most famous example of a normal number is Champernowne's constant 0.123456789101112… Although the definition is very simple, the original proof of normality requires quite some work. In this talk, I present "perfect necklaces", a combinatorial object that yields a simple proof of Champernowne's normality result. And with a class of them, the "nested perfect necklaces", I explain M. Levin's constant, the number with the fastest known speed of convergence to normality.

Abstract: We will talk about characterizations of the local Ramsey property relative to several types of coideal; in different contexts, including infinite games, block sequences of vectors and topological Ramsey spaces.

Abstract: The theory of graph quasirandomness studies several asymptotic properties of the random graph that are equivalent when stated as properties of a deterministic graph sequence and was one of the main motivations for the theory of dense graph limits, also known as theory of graphons. Since the theory of graphons can itself be used to study graph quasirandomness and can be generalized to a theory of dense limits of models of a universal first-order theory, a natural question is whether a general theory of quasirandomness is possible.

In this talk, I will briefly introduce the general theory of dense limits of combinatorial objects (often associated with the name continuous combinatorics) and talk about the notion of natural quasirandomness, a generalization of quasirandomness to the same general setting of universal first-order theories. The main concept explored by our quasirandomness properties is that of unique coupleability that roughly means that any alignment of two limit objects on the same ground set "looks random".

This talk is based on joint work with Alexander A. Razborov.

Abstract: In 1970, Matiyasevich proved that Hilbert’s Tenth Problem over the integers is undecidable, building on work by Davis-Putnam-Robinson. Hilbert’s Tenth Problem over the rationals is still open, but it could be resolved by giving an existential definition of the integers inside the rationals. Proving whether such a definition exists is still out of reach. However, we will show that only “very few” algebraic extensions of the rationals have the property that their ring of integers are existentially or universally definable. Equipping the set of all algebraic extensions of the rationals with a natural topology, we show that only a meager subset has this property. An important tool is a new normal form theorem for existential definitions in such extensions. As a corollary, we construct countably many distinct computable algebraic extensions whose rings of integers are neither existentially nor universally definable. Joint work with Russell Miller, Caleb Springer, and Linda Westrick.

Abstract: I’ll explain how an ultraproduct of polynomial rings in an unbounded number of variables turns out to be surprising, and how this was used to answer some old questions in commutative algebra. This is joint work with Steven Sam and Andrew Snowden.

Abstract: Analogues of the infinite Ramsey theorem to infinite structures have been studied since the 1930’s, when Sierpiński gave a coloring of pairs of rationals into two colors such that, in any subset of the rationals forming a dense linear order, both colors persist. Such a coloring is called “unavoidable” since both colors persist in any infinite substructure isomorphic to the original (in this case the rationals as a linear order). In the 1970’s, Galvin showed that two is the optimum number for pairs of rationals, while Erdős, Hajnal and Pósa extended Sierpiński’s result to colorings of edges in the Rado graph. These results instigated a steady stream of results for the next several decades, a pinnacle of which was the work of Laflamme, Sauer, and Vuksanović finding the exact number of colors for unavoidable colorings of finite graphs inside the Rado graph, as well as other Fraïssé structures with finitely many binary relations, including the generic tournament. This exact number is called the “big Ramsey degree”, a term coined by Kechris, Pestov, and Todorčević.

In this talk, we will provide a brief overview of the area of big Ramsey degrees on Fraïssé limits. Then we will present recent joint work with Rebecca Coulson and Rehana Patel characterizing the big Ramsey degrees for some seemingly disparate Fraïssé classes. We formulate an amalgamation property, which we call the Substructure Free Amalgamation Property, and show that every Fraïssé relational class with finitely many relations satisfying SFAP has big Ramsey degrees which are characterized in a manner as simply as those of the Rado graph. A more general property for disjoint amalgamation classes, which we call SDAP⁺, also ensures the same simple characterization of big Ramsey degrees. One of the novelties of our approach is that we build trees of quantifier-free 1-types with special nodes coding the vertices in a given enumerated Fraïssé limit. Then we use the method of forcing to do an unbounded search for a finite object, which produces in ZFC the exact big Ramsey degrees for these structures. SDAP⁺ holds for unrestricted relational structures, relational structures with forbidden 3-irreducible substructures, and others, producing new lines of results while recovering in a streamlined manner several previous results, including those of Laflamme, Sauer, and Vuksanović.

Abstract: We will survey some recent results in the reverse mathematics of Noetherian algebra, and in particular Noether's Decomposition Lemma, which states that a Noetherian ring has only finitely many minimal prime ideals. Such an analysis naturally leads one to formulate a combinatorial principle, namely, the Tree-Antichain Principle, which states that every binary-branching tree with infinitely many splittings has an infinite antichain.

Abstract: We will explore how the dynamical notion of universal minimal flow interacts with the topological group notion of group extension of or by a compact group. As an application, for a locally compact group of automorphisms of a countable first-order structure with a compact open normal subgroup, we completely describe the underlying space of its universal minimal flow.

Abstract: The infinite Ramsey theorem states that whenever one colors the n-element subsets of the natural numbers using finitely many colors, one can find an infinite subset whose n-element subsets all receive a single color. When considering colorings of finite substructures of more interesting countable structures, the situation becomes more complex. For example, one can color the pairs of the rational linear order using two colors so that on any infinite subset which itself is a rational linear order, both colors of pairs appear. However, two colors is the worst possible; given such a coloring with three colors, one can pass to an infinite, rationally ordered subset where only two colors appear. Combining these results, we say that 2 is the big Ramsey degree of the two-element linear order. In this talk, we give a survey on some recent work involving big Ramsey degrees, culminating in a discussion of recent joint work with Balko, Chodounský, Dobrinen, Hubička, Konečný and Vena which characterizes the precise big Ramsey degrees for many classes of binary Fraïssé structures.

Abstract: The talk will recall some definitions and results on Scott complexity of individual countable structures, Borel complexity of classes of structures, and Borel cardinality, for comparing classification problems for different classes. I will try to indicate how methods from computability may sometimes be used in this connection, even for results that do not mention computability. I will state some results on torsion-free Abelian groups, due to Downey-Montalbán, Hjorth, Thomas, and Paolini-Shelah. Finally, I will mention a project with Turbo Ho and Russell Miller, saying what we would like to do, but have not done.

Abstract: Category theory appears to study "all" objects of a given sort, such as the category of all groups. In ZFC, such categories are proper classes; but this prohibits constructions like functor categories. Thus, category theorists often use instead the category of groups (say) of rank below some inaccessible. But this raises new issues, such as whether arithmetical theorems proven using category theory (e.g. Wiles's proof of Fermat's Last Theorem) are still true without large cardinals. McLarty has argued convincingly that they are, but at the expense of requiring familiar category-theoretic notions to be encoded in unnatural-looking ways.

This situation can be improved by using a "synthetic" language for categories. Instead of using set theory as the implicit foundation for all mathematics, we can regard ordinary category-theoretic proofs as implicitly formalized in a type-theoretic language, which contains an apparently "inaccessible" universe of sets. Then we encapsulate McLarty's encodings (which are essentially a variant of "indexed category theory") in a single theorem interpreting this type-theoretic language into a set theory containing a much weaker universe.

Abstract: Wadge’s comparability lemma says that the Borel sets are almost linearly ordered under Wadge reducibility: For any two Borel sets A and B, either A is a continuous pre-image of B, or B is a continuous pre-image of the complement of A. Wadge’s proof uses Borel determinacy, which is not provable in second-order arithmetic (H. Friedman). Using deep and complex techniques, Louveau and Saint-Raymond showed that Borel Wadge comparability is provable in second-order arithmetic, but did not explore its precise proof-theoretic strength. I will discuss recent work aiming to clarify this.

One of the main technical tools we use is Montalbán’s “true stage” machinery, originally developed for iterated priority constructions in computable structure theory, but more recently used by Day and Marks for their resolution of the decomposability conjecture.

Joint work with Adam Day, Matthew Harrison-Trainor, and Dan Turetsky.

Abstract: I'll talk about a connection between model theory and recursive model theory. In answering Zilber's trichotomy conjecture, Hrushovski built a strongly minimal theory with a geometric property called flatness. Flatness precludes the existence of a definable group, but it does much more. I'll discuss what the assumption of flatness tells us about being able to build recursive models of strongly minimal theories.

The needs of the recursion-theoretic constructions translate into two purely model-theoretic questions. I'll discuss the solution to these questions and try to highlight the connections with recursive model theory.

Joint work with Omer Mermelstein.

Abstract: TBA

Abstract: TBA

Abstract: A natural way to measure the similarity between two infinite binary sequences X and Y is to take the (upper) density of their symmetric difference. This is the Besicovitch distance on Cantor space. If d(X,Y) = 0, then we say that X and Y are coarsely equivalent. Greenberg, Miller, Shen, and Westrick (2018) proved that a binary sequence has effective (Hausdorff) dimension 1 if and only if it is coarsely equivalent to a Martin-Löf random sequence. They went on to determine the best and worst cases for the distance from a dimension t sequence to the nearest dimension s>t sequence. Thus, the difficulty of increasing dimension is understood.

Greenberg et al. also determined the distance from a dimension 1 sequence to the nearest dimension t sequence. But they left open the general problem of reducing dimension, which is made difficult by the fact that the information in a dimension s sequence can be coded (at least somewhat) redundantly. Goh, Miller, Soskova, and Westrick recently gave a complete solution.

I will talk about both the results in the 2018 paper and the more recent work. In particular, I will discuss some of the coding theory behind these results. No previous knowledge of coding theory is assumed.

Abstract: The first model-theoretic tree properties were introduced by Shelah as a by-product of his analysis of forking in stable theories. Since then, other tree properties have appeared and, together, these combinatorial dividing lines (TP, TP₁/SOP₂, TP₂, SOP₁, etc.) serve as the basis for a growing body of research in model theory. I'll survey the work done in this area (and try to justify the idea that it can be understood as an area) by explaining three of the core ingredients in the theory developed so far: generalized indiscernibles, dividing at a generic scale, and amalgamation.

Abstract: Martin's Conjecture is an attempt to make precise the idea that the only natural functions on the Turing degrees are the constant functions, the identity, and transfinite iterates of the Turing jump. The conjecture is typically divided into two parts. Very roughly, the first part states that every natural function on the Turing degrees is either eventually constant or eventually increasing and the second part states that the natural functions which are increasing form a well-order under eventual domination, where the successor operation in this well-order is the Turing jump.

In the 1980's, Slaman and Steel proved that the second part of Martin's Conjecture holds for order-preserving Borel functions. In joint work with Benny Siskind, we prove the complementary result that (assuming analytic determinacy) the first part of the conjecture also holds for order-preserving Borel functions (and under AD, for all order-preserving functions). Our methods also yield several other new results, including an equivalence between the first part of Martin's Conjecture and a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees.

In my talk, I will give an overview of Martin's Conjecture and then describe our new results.

Abstract: We let R be an o-minimal expansion of a group in a language in which Th(R) eliminates quantifiers, and we let C be a valuational cut in R. We show that if nonforking in certain Morley sequences is symmetric, then the theory of R expanded by a predicate for C and a small number of constants eliminates quantifiers. This is a generalization of results on o-minimal fields with convex subrings satisfying some extra conditions such as T-convexity or o-minimality of the residue field. This is joint work with C. F. Ealy.

Abstract: We introduce the into and within set operations in order to construct sets of arbitrary intrinsic density from any Martin-Löf random. We then show that these operations are useful more generally for working with other notions of density as well, in particular, for viewing Church and MWC stochasticity as a form of density.

Abstract: Let M be an ℵ₀-categorical structure and denote by LO(M) the compact space of linear orders on M. We investigate the probability measures on LO(M) invariant under the natural action of the automorphism group of M and prove, under rather general model-theoretic assumptions, that either M has a definable linear order or LO(M) carries a unique invariant measure (which can be easily and explicitly described). For many structures M, the space LO(M) is the universal minimal flow of the group Aut(M), and our work is in part motivated by a general unique ergodicity question of Angel, Kechris, and Lyons in topological dynamics. Our proof uses techniques from model theory, representation theory, and probability theory, but no special knowledge will be assumed in the talk. I will also provide some background and motivation. This is joint work with Colin Jahel.

Abstract: The connection between the effective dimension of sequences and membership in algorithmically random closed subsets of Cantor space was first identified by Diamondstone and Kjos-Hanssen. In this talk, I highlight joint work with Adam Case in which we extend Diamondstone and Kjos-Hanssen's result by identifying a relationship between the effective dimension of a sequence and what we refer to as the degree of intersectability of certain families of random closed sets (also drawing on work by Cenzer and Weber on the intersections of random closed sets).

As we show, (1) the number of relatively random closed sets that can have a non-empty intersection varies depending on the choice of underlying probability measure on the space of closed subsets of Cantor space - this number being the degree of intersectability of a given family of random closed sets - and (2) the effective dimension of a sequence X is inversely proportional to the minimum degree of intersectability of a family of random closed sets, at least one of which contains X as a member. Put more simply, a sequence of lower dimension can only be in random closed sets with more branching, which are thus more intersectable, whereas higher dimension sequences can be in random closed sets with less branching, which are thus less intersectable, and the relationship between these two quantities (that is, effective dimension and degree of intersectability) can be given explicitly.

Abstract: The generic Muchnik degrees, introduced by Schweber, give a way of comparing the computability-theoretic content of uncountable structures. Though obscured slightly by the need for some set-theoretic machinery, I hope to highlight how this notion really gives an easy and natural way to talk about computable structure theory for uncountable structures. I will focus on the tool of complexity profiles.

Complexity profiles are a way of measuring, for two structures A generic Muchnik reducible to B, which subsets of A can be defined using B. The complexity profile of A on itself is the natural analog of considering the relatively intrinsically Σ_ksets in A.

Using complexity profiles, I will compare three generic muchnik degrees: Cantor space < Baire space < the Borel-complete degree. In particular, I will describe some dichotomy theorems regarding simple expansions of these and describe how to build degrees strictly between them. (Joint work with Joseph S. Miller, Noah Schweber, and Mariya Soskova.)

Abstract: Reverse mathematics studies the strength of axioms needed to prove various mathematical theorems. Often, the theorems have the form ∀X ∃Y ψ(X,Y) with X and Y denoting subsets of N and ψ arithmetical, and the logical strength required to prove them is closely related to the difficulty of computing Y given X. In the early decades of reverse mathematics, most of the theorems studied turned out to be equivalent, over a relatively weak base theory, to one of just a few typical axioms, which are themselves linearly ordered in terms of strength. More recently, however, many statements from combinatorics, especially Ramsey theory, have been shown to be pairwise inequivalent or even logically incomparable.

The usual base theory used in reverse mathematics is RCA0, which is intended to correspond roughly to the idea of "computable mathematics". The main two axioms of RCA₀ are: comprehension for computable properties of natural numbers and mathematical induction for c.e. properties. A weaker theory, in which induction for c.e. properties is replaced by induction for computable properties, has also been introduced, but it has received much less attention. In the reverse mathematics literature, this weaker theory is known as RCA^*₀.

In this talk, I will discuss some results concerning the reverse mathematics of combinatorial principles over RCA^*₀. We will focus mostly on Ramsey's theorem and some of its well-known special cases: the chain-antichain principle CAC, the ascending-descending chain principle ADS, and the cohesiveness principle COH.

The results I will talk about are part of a larger project joint with Marta Fiori Carones, Katarzyna Kowalik, Tin Lok Wong, and Keita Yokoyama.

Abstract: We revisit two research programs that were proposed in the 1960's, remained largely dormant for five decades, and then become hot areas of research in the last decade.

The monograph "Continuous Model Theory" by Chang and Keisler, Annals of Mathematics Studies (1966), studied structures with truth values in [0,1], with formulas that had continuous functions as connectives, sup and inf as quantifiers, and equality. In 2008, Ben Yaacov, Berenstein, Henson, and Usvyatsov introduced the model theory of metric structures, where equality is replaced by a metric, and all functions and predicates are required to be uniformly continuous. This has led to an explosion of research with results that closely parallel first-order model theory, with many applications to analysis. In my forthcoming paper "Model Theory for Real-valued Structures", the "Expansion Theorem" allows one to extend many model-theoretic results about metric structures to general [0,1]-valued structures - the structures in the 1966 monograph but without equality.

My paper "Ultrapowers Which are Not Saturated", J. Symbolic Logic 32 (1967), 23-46, introduced a pre-ordering M ≤ N on all first-order structures, that holds if every regular ultrafilter that saturates N saturates M, and suggested using it to classify structures. In the last decade, in a remarkable series of papers, Malliaris and Shelah showed that that pre-ordering gives a rich classification of simple first-order structures. Here, we lay the groundwork for using the analogous pre-ordering to classify [0,1]-valued and metric structures.

Abstract: The ability for a recursively enumerable Turing degree d to bound certain important lattices depends on the degree's fickleness. For instance, d bounds L₇ (or the 1-3-1 lattice) if and only if d's fickleness is > ω (≥ ω^ω, respectively). We work towards finding a lattice that characterizes the > ω² levels of fickleness and seek to understand the challenges faced in finding such a lattice. The candidate lattices considered include those that are generated from three independent points, and upper semilattices that are obtained by removing the meets from important lattices.

Abstract: Independent families on ω are families of infinite sets of integers with the property that for any two disjoint finite subfamilies A and B, the set ⋂ A \ ⋃ B is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at ω (at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalized Baire spaces, which remain wide open.

Abstract: There are many notions of algorithmic randomness in addition to classic Martin-Löf randomness, such as 2-randomness, weak 2-randomness, computable randomness, and Schnorr randomness. For each notion of randomness, we consider the statement "For every set Z, there is a set X that is random relative to Z" as a set-existence principle in second-order arithmetic, and we compare the strengths of these principles. We also show that a well-known characterization of 2-randomness in terms of incompressibility can be proved in RCA, which is non-trivial because it requires avoiding the use of Σ⁰₂-bounding.

This work is joint with André Nies.

Abstract: We show a number of constructions in the theory of partial combinatory algebras which highlight the interplay between topos theory and recursion theory.

Abstract: In earlier work with Knight and Solomon, we bounded the computational complexity of the root-taking process over Puiseux and Hahn series, two kinds of generalized power series. But it is open whether the bounds given are optimal. By looking at the most basic steps in the root-taking process for Hahn series, we together with Hall and Knight became interested in the complexity of problems associated with well-ordered subsets of a fixed ordered abelian group. Here we provide an overview of the results so far in both these settings.

Abstract: Two groups are regarded to be the same if they are isomorphic, two topological spaces are regarded to be the same if they are homeomorphic, two metric spaces are regarded to be the same if they are isometric, two categories are regarded to be the same if they are equivalent, etc. In Voevodsky's Univalent Foundations (HoTT/UF), the above become theorems: we can replace "are regarded to be the same" by "are the same". I will explain how this works. I will not assume previous knowledge of HoTT/UF or type theory.

Abstract: We show that a computable real-valued function f has Luzin's property (N) if and only if it reflects Π¹₁-randomness, if and only if it reflects Δ¹₁-randomness relative to Kleene's O, and if and only if it reflects Kurtz randomness relative to Kleene's O. Here a function f is said to reflect a randomness notion R if whenever f(x) is R-random, then x is R-random as well. If additionally f is known to have bounded variation, then we show f has Luzin's (N) if and only if it reflects weak-2-randomness, and if and only if it reflects Kurtz randomness relative to 0'. This links classical real analysis with algorithmic randomness.

Joint with Arno Pauly and Liang Yu.