The Bulletin of Symbolic 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 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 . 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 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 ? 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 geometry. It remains an important approach in the ongoing pursuit of refinements to this key theorem. In this talk, I will survey a number of results on different types of smooth parameterization in the o-minimal setting, focussing in particular on questions of definability and effectivity, as well as discuss some applications to point counting results.

 LINDA WESTRICK, Borel sets in reverse mathematics. Department of Mathematics, Penn State University, McAllister Building, University Park, PA 16802, USA. E-mail: [email protected]. Reverse mathematics is a program that seeks to answer the question: which set-existence axioms are needed to prove theorems of ordinary mathematics? However, even the simplest theorems about Borel sets—that they have the property of Baire and are measurable—had not been analyzed in a satisfactory way. The standard definition of a Borel set in reverse mathematics has the axiom of arithmetic transfinite recursion (ATR) already baked into it. Thus no interesting reversals could be obtained for theorems such as the above, which are usually proved using only ATR. We propose a new definition for a Borel set in reverse mathematics, that of a completely determined Borel set. Using this definition, we find that “Every Borel set has the property of Baire” and “Every Borel set is measurable” are strictly ATR 1 1 weaker than but do imply the existence of Δ1-generics and Δ1-randoms respectively. The same techniques permit a tighter analysis of the Borel dual Ramsey theorem for 3-partitions, and are beginning to populate a previously uncharted territory of the reverse mathematics universe.

Abstracts of contributed talks

 ALEXI BLOCK GORMAN, Companionability characterization for the expansion of an O- minimal theory by a dense subgroup. Mathematics Department, University of Illinois at Urbana-Champaign, 1409 W Green Street, Urbana, IL 61801, USA. ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING 5

E-mail: [email protected]. Let T be a complete o-minimal theory expanding the theory of groups, and let L be the language in which T admits quantifier elimination. Let TG be the expansion of T by a unary predicate G that picks out a dense and codense subgroup. I provide a full characterization for when TG has a model companion. This result is motivated by questions raised in the recent works [1] and [2] concerning preservation results in model companions for some neostability properties. I restrict my attention to the o-minimal setting because this permits a full and geometric characterization for companionability. I supply examples both in which the predicate is an additive subgroup, and where it is a mutliplicative subgroup. I conclude with a brief discussion of neostability properties, and give examples that illustrate the lack of preservation for properties such as strong, NIP, and NTP2, though there are also examples for which the model companion preserves NTP2. [1] Christian d’Elbee´ , Generic expansions by a reduct, preprint, http://arxiv.org/ abs/1811.06108v2, 2018. [2] Alex Kruckman, Minh Chieu Tran, and Erik Walsberg, Interpolative fusions, preprint, http://arxiv.org/abs/1811.06108v2, 2018.  ELLIOT KAPLAN, An introduction to HT -fields. University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail: [email protected]. In this talk, I will introduce the class of HT -fields. Let T be an o-minimal theory extending the theory of ordered fields and let K be a model of T which is also equipped with a nontrivial derivation x → x making it an H -field (a particularly nice type of ordered differential field). We require that this derivation interact nicely with the o-minimal structure on K. For example, if K is elementarily equivalent to the real exponential field, we require that exp(a) =exp(a)a for all a ∈ K. If these conditions are met, we say that the expansion of K by this derivation is an HT -field.TheclassofH -fields has been thoroughly explored by Aschenbrenner, van den Dries, and van der Hoeven. I will establish some analogues of their results on H -fields for the class of HT -fields.

 JOACHIM MUELLER-THEYS, Metalogical Extensions. Kurpfalzstr. 53, 69 226 Heidelberg, Germany. E-mail: [email protected]. Modern logic developed ingenious systems adequately formalising the positive part of (mathematical) theories. Nevertheless, there are important negative results as well; for instance, the famous fact that the law of commutativity is not a consequence of the axioms of general group theory. Such results cannot be formalised within the known logical systems. This has motivated our extension of these by new formulæ Θ(α), being to express that α is a theorem with respect to some axiom system Σ. Accordingly, a metalogical extension Σ S α is a conservative extension (of some given consequence relation Σ seq φ), entailing (classical) tautologies, modus ponens, replacement, and having the specific properties Σ  φ implies Σ S Θ(φ)andΣ|= φ implies Σ S¬Θ(φ), reflecting the ways theorems and non-theorems are usually established. How can S be realised? We define the minimalist logical calculus QNI for first-order logic α α ∀xφ x → φ t by means of the following. logical axioms. and rules: if is a tautology, ( ) ( )if t is free for x in φ, x = x, φ(x) ∧ y = x → φ(y), Θ( ); α, α →   , φ →   φ →∀x if x ∈ fv(φ), α ↔  ∀xα ↔∀x,Θ(α) ↔ Θ(). Now we can define deducibility as ∗ ∗ QNI-derivability from the Na¨ıt set Σ := Σ ∪{¬Θ(φ): Σ |= φ}:Σ α :iff Σ QNI α, appearing as generalisation and simplification of sentential S13, based on S5 and Σ = ∅, where 2α is but a notational variant of Θ(α). The semantic realisation Σ = α derives from the satisfaction relation M,V =Σ α with 6 ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING

the specific case M,V =Σ Θ() :iff N ,W =Σ  for all N|=Σ,W. Obtain Σα by replacing all Θ(φ)by or ⊥ depending of whether Σ  φ or Σ |= Σ Σ Σ φ.ThenΣ S α ↔  α, whence Σ S α ↔ r α,wherebyrα is the first Θ-free of α, Σ(α),Σ Σ(α) ,... Eventually, Σ S α iff ΣseqrΣα (Buchholz’s Reducibility Theorem), whence it follows that there is one and only one metalogical extension S, implying the Com- pleteness Theorem:Σ α iff Σ = α. By necessitation and its dual, non-necessitation, it can be obtained that Σ  Θ(α) → α, ¬Θ(α) → Θ(¬Θ(α)), Θ(α) ∧ Θ(α → ) → Θ() for any Σ, whence (T), (5), (K) are general consequences, not coinciding with the logical laws anymore. We provided a quantificational version of S5 deductively characterizing the general consequences probably. Relationship and significance of metalogical extensions to modern modal logic are dis- cussed in “The necessity of reforming modern modal logic” (this Bulletin, 25 (2019), p. 272). Formally, so-called autoepistemic logic is closely related to metalogic extensions, which have been developed since 1991, from 1995 together with Wilfried Buchholz, without whom the present state was unthinkable. It is known that sufficiently strong arithmetical theories cannot be represented com- pletely within themselves,viz.Σ 0,butnon Σ ¬ (0). We have recently shown that even Σ ¬ () for all —still not refuting partial self-representability. However,   ↔¬     ↔¬    ↔ Σ 0  (  0 ), but Σ  0 Θ( 0)(unsoundness), whence even Σ 0 ¬ (0) , ¬ Θ 0 ↔¬Θ(0) , i.e., translation and negation of a certain source are deriv- ableatthesametime(immanent inconsistency).

 IOANNIS SOULDATOS, Local Hanf numbers, Kurepa trees, and limit characterizable car- dinals. University of Detroit Mercy, 4001 W. McNichols Ave, Detroit, MI 48221, USA. E-mail: [email protected]. ℵ ℵ < 0 L , About 30 years ago, Shelah conjectured that if 1 2 ,thenany 1 -sentence with ℵ ℵ 0 ℵ models of size 1 also has models of size 2 . He called 1 the local Hanf number below ℵ 2 0 . ℵ ℵ ≤ κ< 0 L , His conjecture is equivalent to the statement “If 1 2 ,thenno 1 -sentence can have model-extistence spectrum [ℵ0,κ]or[ℵ0,κ)”. κ L , ℵ ,κ Call a characterizable cardinal, if there exists an 1 -sentences with spectrum [ 0 ], κ L , and limit characterizable cardinal, if is a limit cardinal and there exists an 1 -sentences with spectrum [ℵ0,κ). Although characterizable cardinals has been studied before, very little is known for limit characterizable cardinals. In the lecture we will present some recent results about limit characterizable cardinals and ℵ ℵ their connection with the local Hanf numbers below 2 1 and 22 0 .

Dima Sinapova Ioannis Souldatos L , [1] , Kurepa trees and spectra of 1 -sentences, sub- mitted, pre-print, https://arxiv.org/abs/1705.05821.

Abstracts of papers submitted by title

 , Substitution of equals for equals. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA. E-mail: [email protected]. Substitution of equals for equals [SEE] is mentioned in the first paragraph of the 1987 Tarski–Givant masterpiece Formalization of set theory without variables [4]. Tarski and Givant present formalizations of comprehensive set theories and number theories using only variable-free “equations” [sc. identities] as in first-grade arithmetic—no connectives, quantifiers, or even variables. ‘(1 + 2) = 3’ is a variable-free identity. ‘(x + y)=(y + x)’ ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING 7

is a variable-containing identity. But the objects denoted by the constants in Tarski–Givant equations are relations, e.g., set-membership and set-identity. They describe their only rule of inference as “the rule familiar from high-school algebra of replacing equals by equals”. Monk’s informative review [3] calls it “substitution of equals for equals”. Despite its name, in SEE applications, equals are not what are being substituted: the rule goes from one sentence to another—not from one fact to another [1]. Sentences do not contain equals [sc. equal things], only names denoting equals. One thing is not put in place of another thing. One name is put in place of another name [1, 2]. Moreover, ‘equal to’ must be understood to mean “is”, “is one and the same thing as”, “is nothing but”—the “is of identity”. In the sense that one half of a line is equal to the other, not every object equal to a given object satisfies every condition the given object satisfies [2]. Both [3] and [4] use ‘is’ and ‘=’ synonymously—sometimes both in the same sentence. This paper discusses mathematical and philosophical treatment of SEE in the last century. [1] John Corcoran, Sentence, proposition, judgment, statement, and fact, Many sides of logic, College Publications, 2009. [2] John Corcoran and Anthony Ramnauth, Equality and identity, this Bulletin, vol. 19 (2013) pp. 255–256. [3] Donald Monk, Review of and Steven Givant, A formalization of set theory without variables, American Mathematical Society, 1987, Bulletin of the American Mathematical Society, (New Series), vol. 20 (1989), pp. 236–239. [4] Alfred Tarski and Steven Givant, Formalization of set theory without variables, American Mathematical Society, 1987.

 JOHN CORCORAN AND IDRIS SAMAWI HAMID, Meanings of sound. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA. E-mail: [email protected]. The five-letter word ‘sound’ serves as a verb, adjective, and noun. Like many other words, it is ambiguous in the sense of having multiple normal meanings (senses). Some writers prefer ‘polysemous’ to the synonymous but more common ‘ambiguous’. Of the various normal senses of ‘sound’, several are vague in the sense of admitting marginal or borderline cases. Words are ambiguous; senses are vague [1, 2]. By means of stipulative definitions, ‘sound’ has accumulated various technical senses in logic all figuratively connected to the concept “healthy”. Ambiguity and vagueness carry over to logical usage. This paper catalogues uses of ‘sound’ in logic. Valid arguments are those whose conclusions are consequences of their premise-sets [3]. Valid arguments with all true premises are sometimes called sound [4]. However, some established logicians use ‘sound’ as a synonymous substitute for ‘valid’ [6] and some use it interchangeably with ‘valid’ [5]; others reserve ‘sound’ for different uses [3, 4, 7]. Some classic texts don’t use ‘sound’ in a logical sense [8]. Besides modifying ‘argument’, ‘sound’ is also used to modify ‘inference’ [5], ‘rule of inference’ [4, 6, 5], ‘reasoning’ [5], ‘method’ (for generating schemata) [7], and ‘step’ [5], to mention prominent examples. Less often, like ‘complete’, it is used for properties that apply to , specifically com- paring their deducibility relations to their consequence relations. A logic is [strongly] sound iff every conclusion deducible from a given premise-set follows from that set [3, 4, 6, 5]. A logic is [weakly] sound iff every conclusion deducible from the empty premise-set follows from that set, i.e., is devoid of information, tautologous [3, 4]. [1] Alonzo Church, Introduction to , Princeton, 1956. [2] John Corcoran, Sentence, proposition, judgment, statement, and fact, Many sides of logic, College Publications, 2009. 8 ASSOCIATION FOR SYMBOLIC LOGIC 2020 WINTER MEETING

[3] , Argumentations and logic, Argumentation, vol. 3 (1989), pp. 17–43. [4] , Three logical theories, Philosophy of Science, vol. 36 (1969), pp. 153–177. [5] E. J. Lemmon, Beginning logic, Hackett, 1965/1978. [6] Benson Mates, Elementary logic, Oxford, 1972. [7] Willard Quine, , Harvard, 1970/1986. [8] Alfred Tarski, Introduction to logic, Dover, 1995.  JOHN CORCORAN AND JOAQUIN MILLER, Tarski’s proof of the law of identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA. E-mail: [email protected]. Tarski’s LEIBNIZ’S LAW [4, Sect. 17, p. 55] is the second-order [2] sentence in variable- enhanced English [3]: For everything x, for everything y : x = y iff x has every property y has and y has every property x has. The Law of Identity, LI, is: Everything is itself. For everything x : x = x. Tarski says LI can be “derived” from Leibniz’s Law. The paragraph following it begins with “PROOF” and ends with “which was to be proved”. This paper analyzes Tarski’s argumentation [3]. Here we only note three errors. The first error was claiming that the following was obtained by a certain application to Leibniz’s Law of the RULE OF SUBSTITUTION, RS, [4, Sect. 15, pp. 47f]. For everything x : x = x iff x has every property x has and x has every property x has. Applying RS to Leibniz’s Law requires deleting ‘for everything x’, then replacing the three free occurrences of ‘x’ with occurrences of an not containing ‘y’, and adjoining universal quantifiers as needed. That would produce a sentence containing ‘for everything y’, for example: For everything y :0=y iff 0haseverypropertyy has and y has every property 0 has. The second error was claiming that from his first conclusion he derived the following by his sentential logic alone. For everything x : x = x iff x has every property x has. For one thing, he applied to certain non-sentences (“sentential functions”) his sentential logic, which applies only to sentences, Tarski’s third error was claiming that he derived the following by sentential logic alone. For everything x : x has every property x has. Tarski’s stated goal in this section was to illustrate “that that there is no essential difference between reasonings in the field of logic and those in mathematics”. We also discuss this proposition. [1] John Corcoran, Argumentations and logic, Argumentation, vol. 3 (1989), pp. 17–43. [2] , Second-order logic, Essays in memory of Alonzo Church, Kluwer, 1998. [3] , Variable-enhanced English and structural ambiguity, this Bulletin, vol. 24 (2018), pp. 524–525. [4] Alfred Tarski, Introduction to logic, Dover, New York, 1995.