The Bulletin of Symbolic Logic Volume xx, Number x, xxxxx xxxx 2020 WINTER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC Colorado Convention Center Denver, CO January 17–18, 2020 Program Committee: Natasha Dobrinen (Chair), Deirdre Haskell, Daniel Roy. All ASL meeting participants are urged to register. The URL for advance registration for the JMM can be found at http://jointmathematicsmeetings.org/meetings/national/jmm2020/2245 reg. ASL members may be interested in events earlier in the JMM including the AMS-MAA Invited Address Different problems, common threads: Computing the difficulty of mathematical problems by Karen Lange, AMS Special Session Algebras and Algorithms organized by Keith Kearnes, Peter Mayr and Agnes Szendrai, AMS-ASL Special Session Choiceless Set Theory and Related Areas organized by Paul Larson and Jindrich Zapletal, AMS-ASL Special Session Logic Facing Outward organized by Karen Lange and Russell Miller. FRIDAY, JANUARY 17 Room 302, Colorado Convention Center Morning 9:00 – 9:50 Invited Lecture: Victoria Gitman (CUNY Graduate Center), Toy multiverses of set theory. 10:00 – 10:50 Invited Lecture: Wesley H. Holliday (University of California, Berkeley), Axiomatizing reasoning about sets: cardinality, mereology, and decisiveness. Afternoon 1:00 – 1:50 Invited Lecture: Alexander S. Kechris (California Institute of Technology), Countable Borel equivalence relations. 2:00 – 2:50 Invited Lecture: Cameron E. Freer (Massachusetts Institute of Technology), Representation theorems for exchangeable structures: a computability theoretic perspective. 3:00 – 4:50 Contributed Talks: see page 2. c xxxx, Association for Symbolic Logic 1079-8986/xx/xxxx-xxxx/$0.00 1 2 ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING SATURDAY, JANUARY 18 Room 302, Colorado Convention Center Morning 9:00 – 9:50 Invited Lecture: Linda Brown Westrick (Penn State University), Borel sets in reverse mathematics. 10:00 – 10:50 Invited Lecture: Margaret E.M. Thomas (Purdue University), Parameterization, o-minimality and counting points. Afternoon 1:00 – 1:50 Invited Lecture: Benjamin Rossman (University of Toronto), Choiceless polynomial time. CONTRIBUTED TALKS Contributed Talks 3:00 – 3:20 Ioannis Souldatos, Local Hanf numbers, Kurepa trees, and limit characterizable cardinals. 3:30 – 3:50 Elliot Kaplan, An introduction to HT-fields. 4:00 – 4:20 AlexiT.BlockGorman,Companionable characterization for the expansion of an o-minimal theory by a dense subgroup. 4:30 – 4:50 Joachim Mueller-Theys Metalogical extensions. Abstracts of invited talks CAMERON E. FREER, Representation theorems for exchangeable structures: a computability-theoretic perspective. Probabilistic Computing Project, Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave. 46-5121, Cambridge, MA 02139, USA. E-mail: [email protected]. URL Address: http://cfreer.org/. Exchangeability and related hypotheses describe the symmetries under which random sequences, arrays, and other structures are invariant. Classical theorems of probability theory due to de Finetti, Aldous, Hoover, Kallenberg, and others characterize the conditional independence that such structures must exhibit, and provide explicit ergodic decompositions. In this talk, we explore the computable content of these theorems, providing both positive and negative results. We also discuss some motivation from the theory of probabilistic programming languages. Joint work with Nathanael Ackerman, Jeremy Avigad, Daniel Roy, and Jason Rute. VICTORIA GITMAN, Toy multiverses of set theory. CUNY Graduate Center, New York, USA. E-mail: [email protected]. URL Address: https://victoriagitman.github.io. Modern set theoretic research has produced a myriad of set-theoretic universes with fun- damentally different properties and structures. Multiversists hold the philosophical position that none of these universes is the true universe of set theory—they all have equal ontological status and populate the set-theoretic multiverse. Hamkins, one of the main proponents of ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING 3 this view, formulated his position via the heuristic Hamkins Multiverse Axioms, which in- clude such radical relativity assertions as that any universe is ill-founded from the perspective of another universe in the multiverse. With Hamkins, we showed that the collection of all countable computably saturated models of ZFC satisfies his axioms. Countable computably saturated models form a unique natural class with a number of desirable model theoretic properties such as existence of truth predicates and automorphisms. Indeed, any collection of models satisfying the Hamkins Multiverse Axioms must be contained within this class. In a joint work with Toby Meadows, Michał Godziszewski, and Kameryn Williams, we explore which weaker versions of the multiverse axioms have ‘toy multiverses’ that are not made up entirely of computably saturated models. WESLEY H. HOLLIDAY, Axiomatizing reasoning about sets: cardinality, mereology, and decisiveness. University of California, Berkeley, USA. E-mail: [email protected]. In this talk, I give three examples of axiomatizing reasoning about sets in special purpose languages. First, I consider reasoning about comparative cardinality: A B if there is an injection from B to A. I add principles to Boolean algebra to axiomatize reasoning not only about Boolean operations but also about . Second, I consider reasoning about the subset relation (“set-theoretic mereology”) in a modal language: 3ϕ is true at a set A if there is a nonempty B ⊆ A such that ϕ is true at B. I discuss the longstanding open problem of giving a recursive axiomatization of the set of validities for finite sets. Finally, I give an example outside of pure mathematics from voting theory: a set A of voters is decisive over candidates x, y if whenever all voters in A prefer x to y, society must rank x above y. I present an axiomatization of reasoning about decisive sets of voters for voting methods satisfying well- known axioms. These examples are meant to illustrate a methodology familiar to modal logicians: to better understand the core principles governing some mathematical concept, try to axiomatize the validities of a lean language with dedicated operators whose semantics is given by the target concepts. This talk is based on the following papers: [1] Yifeng Ding, Matthew Harrison-Trainor, and Wesley H. Holliday, The logic of comparative cardinality, https://escholarship.org/uc/item/2nn3c35x. [2] Wesley H. Holliday, On the modal logic of subset and superset, Studia Logica, vol. 105 (2017), no. 1, pp. 13–35. [3] Wesley H. Holliday and Eric Pacuit, Arrow’s decisive coalitions, Social choice and welfare, forthcoming, https://doi.org/10.1007/s00355-018-1163-z. ALEXANDER S. KECHRIS, Countable Borel equivalence relations. Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected]. The theory of definable equivalence relations has been a very active area of research in descriptive set theory during the last three decades. It serves as a foundation of a theory of complexity of classification problems in mathematics. Another source of motivation for the theory of definable equivalence relations comes from the study of group actions, in a descriptive, topological or measure-theoretic context, where one naturally studies the struc- ture of the equivalence relation whose classes are the orbits of an action and the associated orbit space. An important part of this theory is concerned with the structure of countable Borel equivalence relations, i.e., those Borel equivalence relations all of whose classes are countable. It turns out that these are exactly the equivalence relations that are generated by Borel actions of countable discrete groups and this brings into this subject important connections with group theory, dynamical systems, and operator algebras. In this talk, I will give an introduction to some aspects of the theory of countable Borel equivalence relations. 4 ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING BENJAMIN ROSSMAN, Choiceless polynomial time. E-mail:. The “choiceless” computation model of Blass, Gurevich and Shelah (1999, 2002) is an algorithmic framework for computing isomorphism-invariant properties of unordered struc- tures. Machines in this model lack the ability to make arbitrary choices (such as selecting the “first” neighbor of a vertex in a graph), but have the power of parallel execution over all choices from a set. In this talk, I will introduce the choiceless model and discuss an intriguing open question: Is every polynomial-time graph property is computable by a choice- less polynomial-time algorithm? Recent results have demonstrated the surprising power of choiceless algorithms (including formulas of fixed-point logic) to solve problems like match- ing, determinant, linear programming, and isomorphism of Cai-Furer-Immerman graphs. On the other hand, it can be shown that the dual space V ∗ of a finite vector space V is not constructible in choiceless polynomial-time, though lower bounds for decision problems remain elusive. MARGARET E. M. THOMAS, Parameterization, o-minimality and counting points. Department of Mathematics, Purdue University, N. University St., West Lafayette, IN 47907- 2067, USA. E-mail: [email protected]. The geometric tool of smooth parameterization has important diophantine consequences, and was central to the proof of the Pila–Wilkie Theorem connecting o-minimality and dio- phantine