The conference will take place from Monday 7 October to Wednesday 9 October. The event “150 Years Set Theory” and the Zermelo Ring Ceremony will be on Monday afternoon. After the official end of the conference, Hugh Woodin and Jan von Plato will deliver the Young Academy Distinguished Lecture.
The detailed schedule can be found here:
The question if mathematical concepts can be defective in the sense discussed in the conceptual engineering debate, seems to be a polarising one. One the one hand, the high degree of formalisation and exactness in mathematics seems to make its concepts impervious against defects. On the other hand, case studies from mathematical practice have shown that vagueness and inconsistencies do occur. In this talk I take an intermediate position: While it can be shown that mathematical concepts cannot be defective in an often discussed manner, they can be defective under a slightly adapted view. This adaptation highlights the differences between concepts of the formal and concepts of the empirical sciences and therefore support a moderately exceptionalist view about mathematical concepts.
More than 100 years ago, Emil Post characterized the expressive power of sets of Boolean logical connectives; as we say nowadays, he described all Boolean clones. Almost 50 years ago, Thomas Schaefer characterized the computational complexity of deciding primitive positive sentences over Boolean relational structures, the Boolean constraint satisfaction problems. 25 years ago, a tight connection between these two results was discovered and later further links found: to the first preservation theorem in logic, Birkhoff’s HSP theorem, and more recently to finitary endofunctors of the category of sets, the minions. I will talk about parts of this story and a recent joint work with Z. Brady, F. Jankovec, A. Vucaj, and D. Zhuk on expressive power of three-valued logics.
We show that superpolynomial circuit lower bounds for nondeterministic exponential time are consistent with a relatively strong theory of bounded arithmetic. Additionally, we establish a magnification result on the hardness of proving circuit lower bounds.
Formulating a theory of quantum gravity is one of the biggest open problems in mathematical physics. Some of the core technical and epistemological difficulties come from the fact that General Relativity (GR) is ‘generally covariant’, i.e. invariant under change of coordinates by the arbitrary diffeomorphism of the ambient manifold. The Problem of Observables is a famous instance of the difficulties that general covariance brings into quantization: no non-trivial diffeomorphism-invariant quantity has ever been reported on the collection of all spacetimes. It turns out that there is a good reason for this. In this talk, I will present my recent joint work with Marios Christodoulou and George Sparling, where we employ methods from Descriptive Set Theory in order to show that, even in the space of all vacuum solutions, no complete observables for full GR can be Borel definable. That is, the problem of observables is to ‘analysis’ what the Delian problem is to ‘straightedge and compass’.
To explain phenomena in the world is a central human activity and one of the main goals of rational inquiry. There are several types of explanation: one can explain by drawing an analogy, as one can explain by dwelling on the causes (see e.g. see Woodward (2004)). Amongst these different kinds of explanation, in the last decade philosophers have become receptive to those explanations which explain by providing the reasons or the grounds why a statement is true; these explanations are often called conceptual explanations (e.g. see Betti (2010)). The main aim of the talk is to propose a logical account of conceptual explanations, developed using the resources of proof theory.
Model theory is a branch of mathematical logic that studies structures like graphs, groups, and fields through formal logical languages, particularly first-order logic. It allows results to be transferred between different structures and identifies common features across various mathematical frameworks, thereby providing a unified approach to algebraic, analytic, differ- ential, and difference geometry. A key goal in model theory is distinguishing between tame structures (e.g. the complex field) and wild structures (e.g. the ring of integers). This distinction was introduced by Shelah in the 1970s (Shelah’s Classification Theory) and is based on restricting combinatorial pattern given by definable binary relations. At the apex are the stable structures, such as the complex field, with two complementary extensions: NIP structures (including p-adic fields) and simple structures (such as pseudofinite fields).
In this talk, we briefly discuss fields whose theory is tame in the above sense. Afterwards, we leave the binary world, and introduce “n-ary” Classification theory. Instead of “controlling” 2-ary relation, we only assume that n-ary relations are tame. We give examples and disscuss groups and fields within these classes of theories.
There are many “paradoxical sets” of reals that can be obtained using a well-ordering of the reals, which is a consequence of the Axiom of Choice. In ZF, can we recover the well-ordering of the reals from the existence of a given paradoxical set? Under certain conditions of Extendability and Amalgamation, we give a negative answer to this question. In particular, we solve it for the paradoxical set given by a partition of R³ in unit circles.
The interplay between the axiom of determinacy and inner models with large cardinals is one of the central topics in set theory. Recent developments in inner model theory, in particular the theory of hod mice, provide us with the way to construct various models that satisfy strong forms of determinacy. In this talk I’ll talk about my contribution in this direction over the last few years.
Many believe mathematical truth is indisputable. However, the set-theoretic independence phenomenon challenges this idea. Certain statements about infinite sets, like the continuum hypothesis, are neither true nor false according to the standard axioms.
While philosophers have offered various diagnoses of this problem, my PhD thesis posits that the set-theoretic community is key to solving the issue, proposing a pragmatic approach. It presents the first extensive empirical study, featuring interviews with 28 set theorists from varied backgrounds. It explores the spectrum of disagreement and agreement among practitioners, predicting whether the community will adopt new axioms.
Overall, my PhD thesis demonstrates the potential of empirical research for philosophical purposes, also contributing to debates in social epistemology. It addresses a complex problem, applies a novel sociological method, and reveals much about the inner workings of a scientific community in the foundations of mathematics.
The nonstationary ideal on ω_1 is ω_1-dense if there is a set B of ω_1-many stationary subsets of ω_1 that every stationary subset of ω_1 contains an element of S on a club. We show that this principle holds true in a stationary set preserving forcing extension, assuming a supercompact limit of supercompact cardinals exists. This answers a question of Woodin positively.
I give an overview of my PhD thesis which is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators. For this, the current logical methodology of proof mining is extended by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets. The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.
The thesis examines the dynamics of formalizations in mathematical practices from various perspectives. This includes a Wittgensteinian analysis, the study of the role of real-world abstractions in theory building, the role of prototypical examples through a Frame-Semantical approach, questions concerning the narratives or prose surrounding mathematical practice, and methodological considerations, such as the role empirical work plays in the study of mathematical practices.
I consider notions of definability, and how they classify the complexity of sets and structures. In my thesis, I considered two questions in classical mathematical subjects (group theory and geometric measure theory) from the lens of computability theory and set theory. This perspective has led to theorems of interest to logicians and non-logicians alike. I will outline the ideas behind these definability approaches, explain the connections to classical mathematics, and put our novel results into context.
A directed graph (digraph) G can primitively positively construct another digraph H if H is homomorphically equivalent to a first order interpretation of G that only uses primitive positive formulas. In this way, we define the poset of all finite digraphs ordered by primitive positive constructability. This poset, which is the object I studied in my dissertation, is important for studying the complexity of constraint satisfaction problems and for studying clones ordered by minion homomorphisms. In this talk I will introduce the poset and present some results from my dissertation. In particular, I will present the three topmost levels of the poset and I will describe the subposet consisting of digraphs without sources or sinks.
In 1875, du Bois Reymond introduced a flexible type of mathematical infinity: growth rates of “sufficiently regular” real-valued functions. He showed that certain gaps between sequences of regular growth rates could be filled by other regular growth rates. His (and Hardy’s) results were recently vastly improved by Aschenbrenner, van den Dries and van der Hoeven showed that any gap between sequences of growth rates forming a differential field be filled inside a larger differential field.
One hundred years after du Bois Reymond’s seminal work, Conway defined surreal numbers by recursively creating and filling gaps in the set-theoretic universe. It turns out that they can be identified in a canonical way with regular growth rates… so long as one allows for a more general notion of growth rate called a hyperseries. Hyperseries are formal series involving exponentials and logarithms of a positive infinite variable, as well as “transfinite iterates” of the exponential and logarithm.
I will explain how to represent surreal numbers as hyperseries. Time allowing, I will present ongoing work to let hyperseries act as monotonous, infinitely differentiable functions on the class of surreal numbers.
This is based on joint work with Joris van der Hoeven, Elliot Kaplan and Vincenzo Mantova.
Epistemic logic has been successfully applied to distributed systems, describing agents’ epistemic attitudes given the current set of alternatives the agents consider possible. Dynamic epistemic logic describes how these epistemic attitudes change based on the new information agents receive. In a broader philosophical view, this appeals to an a posteriori kind of reasoning, where agents update the set of alternatives considered possible based on their “experiences”. Usually, this set is provided to agents prior to system execution by a system designer, setting the arena in which epistemic reasoning takes place and thus constituting the static body of a priori beliefs of agents. While fault-tolerant systems are designed in order to provide some resilience to faults like e.g., byzantine faults, the lack of autonomous a priori dynamics prohibits agents to autonomously adapt to situations arising during execution not envisioned by the system designer, because of e.g., design mistakes, offloading the responsibility of the system’s correctness solely on the direct intervention of the system designer. With systems becoming more and more complex and large, the task of fixing design errors becomes prohibitively costly, invoking the need to allow agents to fix such mistakes “on the fly” such as in the increasingly popular self-adaptive and self-organizing (SASO) systems. Rather than just updating agents’ a posteriori beliefs, this requires modifying their a priori beliefs about the system’s global design and parameters. We present a formalization of such a priori updates, enriching standard epistemic semantics with updates capturing the self-correcting process initiated by agents in case of unexpected behavior. The overall goal of this kind of update, which is triggered by an agent’s detection of some unforeseen situations, is to adapt the set of states that must be considered possible by agents in order to explain the unexpected behavior, restoring correctness.
Abstractionist theories in philosophy of mathematics are systems composed by a logical theory augmented with an abstraction principle (AP), of the form: ∀X∀Y (@X = @Y ) ↔ E(X, Y ), originally proposed in order to pursue the Fregean project of a logicist foundation of arithmetic. In general, any abstractionist version of Logicism turns out to be affected by issues concerning its consistency and its logicality (cf. [5], [4], [1], [3], [2]).
In this talk, on the one side, I prove that any consistent revision of the crucial axiom BLV turns out to be logical (i.e. permutation invariant) if we substitute the so-called Canonical interpretation function with an arbitrary one (cf. [2]); on the other side, I show that such an arbitrary interpretation allows us to identify a restriction of BLV (W-BLV: ∀F ∀G(εF = εG ↔ ∀x(F x ↔ Gx)∧ε(π(F )) = π(εF ))) that, if combined with a weakening of classical logical background in a (negative) free logical one, is able to precisely exclude the paradoxical concepts and, at the same time, to derive second-order Peano axioms. This means that this system – that we’ll call Arbitrary Frege Arithmetic – is able to recover both Frege’s goals of consistency and logicality.
References
[1] Antonelli, G. A. (2010). Notions of invariance for abstraction principles. Philosophia Mathematica, 18(3), 276-292.
[2] Boccuni, F., Woods, J. (2018). Structuralist neologicism. Philosophia Mathematica, 28(3), 296-316.
[3] Cook, R. (2016).Abstraction and Four Kinds of Invariance (Or: What’s So Logical About Counting), Philosophia Mathematica, 25,1, 3–25.
[4] Fine, K. (2002). The limits of abstraction. Clarendon Press.
[5] Tarski, A. (1956). The concept of truth in formalized languages. Logic, semantics, metamathematics, 2(152-278), 7.
In a recent publication, Çevik (2023) introduces the hierarchical multiverse of sets which organises different set theories by assigning a degree of intentionality to each sentences φ independent of ZFC. This approach allows Çevik to score the universes in the multiverse in such a way that the resulting picture is still compatible with the multiverse perspective, but induces an order making the multiverse hierarchical. We claim that this comes down to choosing between φ and ¬φ, depending on which has the higher degree of intensionality. This, however, has the (unintended) consequence of approximating a maximally consistent theory T* := ZFC ∪ {φ | dg(φ) > dg(¬φ)} ∪ {¬φ | dg(¬φ) > dg(φ)} containing ZFC plus all sentences with the hig hest degree of intentionality. This talk will critically examine the implications of this approach, arguing that it implicitly commits Çevik to a view of the universe that assumes a single, coherent mathematical reality. This commitment seems to conflict with his “pluralist dilemma”, which posits that one must choose between liberalising mathematical ontology and maximising the determinacy of mathematical truth, but cannot fully achieve both. We argue that the tension between hierarchical pluralism and the pluralist dilemma merely points to the old philosophical problem of how to justify the adoption of one set theory over the other.
A synonymy relation for formal theories attempts to capture their equivalence in meaning. I will discuss some intuitively plausible principles of theoretical synonymy that arise from an inferential concept of meaning. It is shown that relations of theoretical synonymy often used in logic, such as definitional equivalence and bi-interpretability, do not fulfill these principles. For this purpose, simple toy theories are used as well as mathematically interesting examples from geometry and field theory. As a result, I will examine to what extent the initially motivated principles are suitable for axiomatizing a more adequate relation of synonymy that can be used in logic and philosophy.
In this talk we present joint work by G. Cignarale, S. Felber, and H. Rincón Galeana, introducing a sufficient epistemic condition to solve stabilizing agreement, the non-terminating variant of the well known consensus problem. In stabilizing agreement (introduced as a variant of the famous Byzantine Generals Problem) agents are given (possibly different) initial values, with the goal to eventually always decide on the same value. While agents are allowed to change their decisions finitely often, they are required to agree on the same value eventually, but crucially need not know when. We capture these properties in temporal epistemic logic and we use the Runs and Systems framework to formally reason about stabilizing agreement problems. We then epistemically formalize the conditions for solving stabilizing agreement and identify sufficient knowledge that the agents need to acquire during any execution to satisfy these conditions.
The tilting construction (as introduced by Fontaine) provides a way to transfer theorems between the worlds of characteristic zero and positive characteristic. Classically, this was done for perfectoid fields: for each perfectoid field of characteristic zero, we can obtain its tilt – a perfectoid field of positive characteristic. Perfectoid fields are complete non-discretely valued fields of rank 1 that satisfy some perfectness condition. In my talk, I will tell you how we can extend the tilting construction to certain valued fields of higher rank using model theory. The model-theoretic tilt we obtain is only defined up to elementary equivalence, so we tilt the theories rather than the fields themselves.
Proof mining is a subfield of applied proof theory concerned with the extraction, with the help of proof-theoretic tools, of new quantitative and qualitative information from mathematical proofs. This paradigm of research, developed by Ulrich Kohlenbach beginning with the 1990s, is inspired by Kreisel’s program on unwinding of proofs from the 1950s. In this talk we present recent applications of proof mining to asymptotic regularity proofs for Halpern-type nonlinear iterations. General logical metatheorems from proof mining guarantee that one can obtain effective uniform rates of asymptotic regularity for these iterations. Asymptotic regularity is a very important notion in nonlinear analysis and optimization, introduced by Browder and Petryshyn in the 1960s for the Picard iteration. This is joint work with Paulo Firmino.
One way to study the properties of the infinite cardinals is to examine the extent to which they can be changed by forcing. Around 1970, Bukovský and Namba independently showed that ℵ_2 can be forced to be an ordinal of cofinality ℵ_0 without collapsing ℵ_1. The forcings they used and their variants are now known as Namba forcing, and are notable for quasi-minimality properties: In particular, Namba forcing has no subforcings collapsing ℵ_2, unlike the Lévy collapse. Also unlike the Lévy collapse, Namba forcing always collapses ℵ_3 to have cardinality ℵ_1, as is demonstrated by work of Shelah. One can then wonder what interplay there is between minimality and collapses. In a 1990 paper, Bukovský and Copláková asked whether there can be an extension that collapses ℵ_2 to have cardinality ℵ_1 in a minimal way without collapsing ℵ_3. Using a measurable cardinal, we show that it is in fact possible to construct such an extension.
Although o-minimality is a theory of ordered structures, many theorems in complex analysis can be recovered in the o-minimal context. Applications of o-minimality to number theory and functional transcendence often rely on the theory of o-minimal complex analysis developed by Peterzil and Starchenko. In this talk, I will present joint work with P. Speissegger in which we prove that the Gamma function and Riemann zeta function are o-minimal on certain unbounded complex domains. I will also discuss potential applications of this result.
Undoubtedly, one of the most complicated instances of proof mining to date is the proof-theoretical unwinding of Reich’s theorem, one of the most pivotal results in functional analysis, carried out in [U. Kohlenbach and A. Sipos, “The finitary content of sunny nonexpansive retractions”, Communications in Contemporary Mathematics, 23(1):1950093, 63 pp., 2021].
In this talk, we introduce the notion of a nonlinear smooth space, generalizing both CAT(0) spaces and smooth Banach spaces. We discuss how this notion allows for a unified treatment of several mathematical proofs in functional analysis and is suitable for proof mining metatheorems on the extraction of bounds. In particular, we show that Kohlenbach’s and Sipos’s treatment of Reich’s result can be appropriately discussed in this nonlinear setting.
Transseries emerged in connection with Écalle’s work on Dulac’s problem and Dahn and Göring’s work on nonstandard models of real exponentiation, and some can be viewed as asymptotic expansions of solutions to differential equations. More recently, Aschenbrenner, Van den Dries, and Van der Hoeven completely axiomatized the elementary theory of the differential field of (logarithmic-exponential) transseries and showed that it is model complete. This talk concerns pairs of models of this theory such that one is a tame substructure of the other in a certain sense. I will describe the model theory of such transserial tame pairs, including a model completeness result for them, which can be viewed as a strengthening of the model completeness of large elementary extensions of the differential field of transseries, such as hyperseries, surreal numbers, or maximal Hardy fields.
Simplicial complexes are a convenient semantic primitive to reason about processes (agents) communicating with each other in synchronous and asynchronous computation. Impure simplicial complexes distinguish active processes from crashed ones, in other words, agents that are alive from agents that are dead. In my talk, I will discuss the three-valued epistemic semantics where, in addition to the usual values true and false, the third value stands for undefined: the knowledge of dead agents is undefined and so are the propositional variables describing their local state. The propositional base of this three-valued semantics is known as Paraconsistent Weak Kleene logic.
We show that if E is a countable Borel equivalence relation, then there is a closed set C of positive Hausdorff dimension such that E is smooth when restricted to C. This result essentially shows that Hausdorff dimension is not a suitable technique to produce “interesting” examples of countable Borel equivalence relations, matching other recent results in the area. Joint work with Andrew Marks and Ted Slaman.
In my talk I will explain the notion of a (maximal) cofinitary group (mcg) and its corresponding cardinal characteristic a_g. Then, I will present some recent research regarding forcing arguments and definability of such mcg’s. In particular, I will introduce a notion of tightness for mcg’s which implies the forcing indestructibility for various types of tree forcings. Together with a new coding technique for mcg’s we establish the relative consistency of a_g = d < c = ℵ_2 alongside the existence of a ∆^1_3-wellorder of the reals and a co-analytic witness for a_g.
This is joint work with Vera Fischer and David Schrittesser.
New proof mining results on Halpern’s iteration in nonlinear spaces Abstract: We generalize the convergence proof for the Halpern iteration due to Hong-Kun Xu (using his own convergence conditions) and its proof mining analysis due to Daniel Koernlein to the new class of nonlinear spaces (“uniformly smooth hyperbolic spaces”) recently introduced by Pedro Pinto as a common generalization of CAT(0) spaces and uniformly smooth Banach spaces. To our knowledge, the qualitative result here is new even for the class of CAT(0) spaces.
Bourbaki’s 1950 article “The Architecture of Mathematics”, which famously identifies mathematics with the study of particular (mathematical) structures, is seen as a precursor of contemporary mathematical structuralism.
At the same time, fully thriving on the “architectural” metaphor, Bourbaki pointed out that mathematical structures are mutually and hierarchically entangled (he distinguished between mother-, multiple and particular structures), thus also potentially suggesting the existence of a unified foundational framework where more determinate structures (and axioms) are derived from more general (and less determinate) ones.
In the paper, I argue that contemporary set theory, with its abundance of structures (models) all resembling one fundamental type of mother-structure (the universe of sets), and construed as a unified axiomatic theory of the set-theoretic multiverse of ZFC and large cardinals (along the lines of Steel 2014’s MV theory), can be seen as an expression of Bourbaki’s “architectural” (and foundational) ideal for mathematics.
I will then proceed to examine the fruitfulness of this approach for other issues in the philosophy of set theory, such as “multiverse structuralism”, the relationship between mathematical objects and theories, the need for a unified (and ultimate) axiomatic framework.