We show that countable set theory, $ZFC^{-}+\forall x\ |x|\leqω$, is unable to eliminate imaginaries. In other words, this theory cannot provide representatives for arbitrary definable equivalence relations. We also see that $ZFC^{-}$ and ZFC^{-}+\existsκ(Inacc(κ)\wedge\forall x\ |x|\leqκ)$ also fail to eliminate imaginaries.

Set theorists often claim that natural theories are well-ordered by their consistency strength. We call this claim the Consistency Hierarchy Thesis. The goal of this paper is to unpack the philosophical and mathematical significance of this thesis; and to develop an understanding of how it is defended and, more particularly, how one might refute it. We shall see that the thesis involves a curious admixture of mathematics and philosophy that makes it difficult to pin down. We investigate some intriguing attempts to refute the thesis that are hampered by the problem of understanding what makes a theory natural. We then develop a thought experiment exploring the idea of what the ideal scenario for refutation would look like. And we show that a counterexample is impossible if we insist that the counterexample uses respectable (i.e., transitive) models. Finally, we reflect on how these hurdles affect our understanding of the significance of the thesis by drawing a parallel with a more famous claim: the Church-Turing thesis.

This article offers a philosophical overview and investigation of the problem of incompleteness in set theory and what this entails for the ensuing debates about proposed extensions of $ZFC$ . The incompleteness of $ZFC$ is well-known and leaves us with a rich array of competing extensions. What should we make of disagreements between them? We start by considering second-order logic and its categoricity theorems and how they might be used to compare different set theories. We then aim to use interpretability as a way of understanding that some of these debates are insubstantial. This culminates in some discussion of the relationship between interpretability and the generic multiverse. The second half of the article then takes up a more modest goal: we search for common ground and settle for partial agreement between set theories in much the same way that physicists are often content with empirical agreement. We then aim to describe a natural bound on the amount of agreement that we can expect to obtain between reasonable extensions of $ZFC$ .

Halvorson has proposed an intriguing example of a pair of theories whose categories are equivalent but which are not themselves definitionally equivalent. Moreover, it seems obvious that these theories are not equivalent in any intuitive sense. We offer a new topological proof that these theories are not definitionally equivalent. However, the underlying theorem for this claim has a converse that shows a surprising collection of theories, which are superficially similar to those in Halvorson’s example, turn out to be definitionally equivalent after all. This offers some new insight into what is going “wrong” in the Halvorson example.

John Steel's theory, MV, of the generic multiverse provides a foundation for mathematics that aims to neutralize the effects of incompleteness brought on by forcing arguments. Jouko Väänänen's development of internal categoricity arguments provides opportunities to argue that the subject matter of some theory is, in some sense, determined. This paper investigates whether MV is internally categorical.

In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.

This paper assembles a unifying framework encompassing a wide variety of mathematical instruments used to compare different theories. The main theme will be the idea that theory comparison techniques are most easily grasped and organized through the lens of category theory. The paper develops a table of different equivalence relations between theories and then answers many of the questions about how those equivalence relations are themselves related to each other. We show that Morita equivalence fits into this framework and provide answers to questions left open in Barrett and Halvorson [4]. We conclude by setting up a diagram of known relationships and leave open some questions for future work.

The purpose of this paper is to propose and explore a general framework within which a wide variety of model construction techniques from contemporary set theory can be subsumed. Taking our inspiration from presheaf constructions in category theory and Boolean ultrapowers, we will show that generic extensions, ultrapowers, extenders and generic ultrapowers can be construed as examples of a single model construction technique. In particular, we will show that Łoś's theorem can be construed as a specific case of Cohen's truth lemma, and we isolate the weakest conditions a filter must satisfy in order for the truth lemma to work.

Consistency, interpretability and probability are three key instruments in the mathematical philosopher’s kit when it comes to questions of foundational theory comparison. This paper aims to bring these tools together with a focus on theories capable of providing foundations for mathematics with a particular emphasis on set theory. A number of counterintuitive results emerge which are then addressed by offering a novel framework based on what we call pointwise interpretability. We then investigate a plausible, existing instance of this framework, the generic multiverse, and demonstrate that it can be naturally situated within our pointwise interpretability framework.

This paper critically examines two arguments against the generic multiverse, both of which are due to W. Hugh Woodin. Versions of the first argument have appeared a number of times in print, while the second argument is relatively novel. We shall investigate these arguments through the lens of two different attitudes one may take toward the methodology and metaphysics of set theory; and we shall observe that the impact of these arguments depends significantly on which of these attitudes is upheld. Our examination of the second argument involves the development of a new (inner) model for Steel's multiverse theory, which is delivered in the Appendix.

Interpretation is commonly used in mathematical logic to compare different theories and identify cases where two theories are for almost all intents and purposes the same. Similar techniques are used in the comparison between alternative logics although the links between these approaches are not transparent. This paper generalizes theoretical comparison techniques to the case of logical comparison using an extremely general approach to semantics that provides a very generous playing field upon which to make our comparisons. In particular, we aim to develop the useful idea that interpretations should determine inner models.

This paper explores the idea that Descartes’ cogito is a kind of diagonal argument. Using tools from modal logic, it reviews some historical antecedents of this idea from Slezak and Boos and culminates in an orginal result classifying the exact structure of belief frames capable of supporting diagonal arguments and our reconstruction of the cogito.

s of the invited talks and the contributed talks by members of the Association for Symbolic Logic follow. For the Program Committee Denis R. Hirschfeldt Abstract of invited tutorialof invited tutorial ◮ HECTOR PASTEN, Hilbert’s tenth problem beyond the integers. Department of Mathematics, Pontificia Universidad CatÓlica, 4860 Avda. Vicuna Mackenna, Santiago, Chile. E-mail: hector.pasten@mat.uc.cl Hilbert’s tenth problem asked for an algorithm to decide solvability of diophantine equations over the integers. Theworks ofDavis, Putnam,Robinson andMatiyasevich showed that the requested algorithm does not exist. However, the analogous problem remains open in a number of important cases such as the field of rational numbers, rings of integers of number fields, and the ring of complex entire functions. This tutorial aims to present some of the main techniques in this very active area of research. ◮ JOUKO VÄÄNÄNEN, A tutorial on the logic of dependence and independence. Department of Mathematics And Statistics, University Of Helsinki, Helsinki, Finland. E-mail: jouko.vaananen@helsinki.fi URL: http://www.math.helsinki.fi/logic/people/jouko.vaananen/ Tarski defined the semantics of first order logic by giving an inductive definition of what it means for an assignment s of values to variables x1, ...xn to satisfy a formula φ (x1, ... ,xn) in a given model. This definition has served logic well but it is not suitable for expressing dependences and independences between the variables x1, ...xn. One assignment s does not contain enough information to give grounds to make the conclusion that under this assignment e.g., x1 is totally determined by x2, or x5 is totally independent of x7. The situation can be remedied by considering the satisfaction of a formula under not a single assignment, but a whole set of assignments. Such sets play a central role in dependence logic [1] and are called teams to emphasise their collective contribution to truth. In the first lecture of this tutorial the elements of semantics based on teams are presented. Illuminating examples from database theory, imperfect information games, and partially ordered quantifiers are reviewed. The fundamental relationship of dependence logic to existential second order logic, fixpoint logic, non-deterministic polynomial time (NP), and polynomial time (P) is established. In the second lecture the framework developed is applied to social choice theory, quantum information theory, and biology, all major sources of examples of manifestation of dependence and independence phenomena. [1] J. Väänänen, Dependence Logic, Cambridge University Press, 2007. ◮ ANNA ZAMANSKY, Paraconsistent Logics: a Tutorial. Information Systems Department, University of Haifa, Haifa, Israel. E-mail: annazam@is.haifa.ac.il Perhaps the most counterintuitive property of classical logic (as well as of its most famous rival, intuitionistic logic) is the fact that it allows the inference of any proposition from a 174 2020 NORTH AMERICAN ANNUALMEETING single pair of contradicting statements. A lot of work and efforts have been devoted over the years to develop alternatives to classical logic that do not have this drawback. Those alternatives are nowadays called ‘paraconsistent systems’, and the corresponding research area—paraconsistent reasoning. This tutorial, based on a recently published book [1], aims to provide a methodological overview of the richmathematical theory that exists by now concerning themost fundamental part of paraconsistent reasoning: propositional (monotonic) logics. Among those logics we will focus on those which are effective (in the sense that they are decidable, have a concrete semantics, and can be equipped with implementable analytic proof systems). We will start by defining in precise terms basic notions related to paraconsistency. Then we will describe some of the main approaches to paraconsistency: finite-valued semantics (both truth-functional and nondeterministic), logics of formal inconsistency and paraconsistent logics which are based on modal logics. These logical systems will be discussed both from semantical and proof theoretical points of view, and some of them also characterized in terms of minimality or maximality properties. [1]A.Avron,O.Arieli, andA.Zamansky,Theory of effective propositional paraconsistent logics, Studies in Logic 75, College Publications, 2018. Abstracts of invited plenary lecturess of invited plenary lectures ◮ JEREMY AVIGAD, The mechanization of mathematics. Department of Philosophy and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail: avigad@cmu.edu In computer science, formal methods are used for specifying, developing, and verifying complex hardware and software systems. The word “formal” indicates the use of formal languages to write assertions, define objects, and specify constraints. It also indicates the use of formal semantics, that is, accounts of the meaning of a syntactic expression, which can be used to specify the desired behavior of a system or the properties of an object sought. For example, an algorithm may be expected to return a tuple of numbers satisfying a given constraint, expressed in some specified language, whereby the logical account spells out what it means for an object to satisfy the symbolically expressed constraint. Finally, the word “formal” suggests the use of formal rules of inference, which can be used to verify claims or guide a search. Such methods hold great promise for mathematical discovery and verification of mathematics as well. In this talk, I will survey some applications, including verifying mathematical proofs, verifying the correctness of mathematical computation, searching for mathematical objects, and storing and communicating mathematical results. [1] J. Avigad, The mechanization of mathematics, Notices of the American Mathematical Society, vol. 65(2018), no. 6, pp. 681–690, reprinted in The Best Writing on Mathematics 2019, Mircea Pitici, editor, Princeton University Press, Princeton, New Jersey, 2019, pp. 150–170. ◮ OMER BEN-NERIA, Approximating the set theoretic universe by canonical inner models. Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel. E-mail: omer.bn@mail.huji.ac.il The goal of the talk is to introduce the concept of approximating the set theoretic universe V using canonical inner models. We will describe several ways by which a canonical inner model M of V can approximate V , as well as applications and limitations of having such approximations. Our presentation will focus on the inner models L (the constructible universe) andHOD (hereditarily ordinal definable sets), and lead to recent results concerning Woodin’s HOD-conjecture. This is a joint work with Yair Hayut. 2020 NORTH AMERICAN ANNUALMEETING 175 ◮ RINA DECHTER, Reasoning with deterministic and probabilistic graphical models. Donald Bren School of Information and Computer Sciences, UC Irvine, Irvine, CA, USA. “An important component of human problem-solving expertise is the ability to use knowledge about solving easy problems to guide the solution of difficult ones.”—Minsky A longstanding intuition in AI is that intelligent agents should be able to use solutions to easy problems to solve hard problems. This has often been termed the “tractable island paradigm.” How do we act on this intuition in the domain of probabilistic reasoning? This talk will describe the status of reasoning algorithms that are driven by the tractable islands paradigm when solving satisfaction, optimization and likelihood queries described over mixtures of deteministic (logic-based) and probabilistic graphical models. I will show how heuristics generated via variational relaxation into tractable structures, can guide heuristic search and Monte-Carlo sampling, yielding anytime solvers that produce approximations with confidence bounds that improve with time, and become exact if enough time is allowed. ◮ H. JEROME KEISLER, Continuous model theory revisited. University of Wisconsin, Madison, WI, USA. E-mail: keisler@math.wisc.edu We revisit two research programs that were proposed in the 1960s, 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 Model Theory for Metric Structures, Ben Yaacov, Bernstein, Henson, and Usvyatsev, LondonMath. Society LectureNote Series, vol. 350 (2008), pp. 315–427, 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. Here we discuss the “Expansion Theorem,” which allows one to extend many model-theoretic results about metric structures to general [0,1]valued structures—the structures in the 1966 monograph without equality (with no uniform continuity requirement). In the paper “Ultrapowers which are not saturated,” Journal of Symbolic Logic, vol. 32 (1967), pp. 23–46, I introduced a pre-ordering M E 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 E gives a rich classification of simple first-order structures. Here, we discuss analogous of E for general [0,1]-valued structures, and for metric structures. ◮ JULIETTE KENNEDY, Tracking the profile of natural language in foundational practice. Department of Mathematics and Statistics, University of Helsinki, P.O. BOX 68 (Gustaf Hällströmin Katu 2B) FI-00014 University of Helsinki, Finland. E-mail: juliette.kennedy@helsinki.fi UR

This paper reconstructs Steel's multiverse project in his 'Godel's program' (Steel, 2014), first by comparing it to those of Hamkins (2012) and Woodin (2011), then by detailed analysis what's presented in Steel's brief text. In particular, we reconstruct his notion of a 'natural' theory, describe his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks CH might suffer from and isolate what it would take to remove it while working within his framework. As our goal is to present as coherent and compelling a philosophical and mathematical story as we can, we allow ourselves to augment Steel's story in places (e.g., in the treatment of Amalgamation) and to depart from it in others (e.g., the removal of 'meaning' from the account). The relevant mathematics is laid out in the appendices.

The Church-Turing Thesis is widely regarded as true, because of evidence that there is only one genuine notion of computation. By contrast, there are nowadays many different formal logics, and different corresponding foundational frameworks. Which ones can deliver a theory of computability? This question sets up a difficult challenge: the meanings of basic mathematical terms (like set, function, and number) are not stable across frameworks. While it is easy to compare what different frameworks say, it is not so easy to compare what they mean. We argue for some minimal conditions that must be met if two frameworks are to be compared; if frameworks are radical enough, comparison becomes hopeless. Our aim is to clarify the dialectical situation in this bourgeoning area of research, shedding light on the nature of non-classical logic and the notion of computation alike.

This book is a delight to read. It glides effortlessly through hundreds of years of mathematics and its philosophy whilst barely breaking a sweat. Equally at home with Euclid and Eudoxus as with Ca...

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe (Halmos, in: Naive set theory, 1960). In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.

This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort to make sense of Cantor's troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday mathematical practice.

Graham Priest has argued that the fruits of classical set theory can be obtained by naive means through a puzzling piece of reasoning often known as the bootstrapping argument (Priest 2006). I will demonstrate that the bootstrapping involved is best understood as viciously circular and thus, that these fruits remain forbidden. The argument has only one rehearsal in print and it is quite subtle. This paper provides reconstruction of the argument based on Priest (2006) and attempts some fixes and alternative construals to get around some elementary problems. Despite these efforts, the argument remains unconvincing.

We provide infinitary proof theories for three common semantic theories of truth: strong Kleene, van Fraassen supervaluation and Cantini supervaluation. The value of these systems is that they provide an easy method of proving simple facts about semantic theories. Moreover we shall show that they also give us a simpler understanding of the computational complexity of these definitions and provide a direct proof that the closure ordinal for Kripke's definition is omega(CK)(1). This work can be understood as an effort to provide a proof-theoretic counterpart to Welch's game-theoretic (Welch, 2009).