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: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

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.