*Friday 30th June*–

*Saturday 1st July 2023*

This Two-Day Logic Meeting begins in the afternoon of Friday 30th June and ends in the late afternoon of Saturday 1 July. It will feature talks from renowned researchers in several branches of logic.

The meeting is funded by the School of Mathematics and EPSRC.

**Confirmed speakers:**** **

**Sam Coskey, University College London**

*Title*: Conjugacy, classification, and complexity

*Abstract*: We investigate the classification of automorphisms of a countable structure up to conjugacy. We aim to identify the complexity of this classification for a variety of structures. To study the complexity, we use the Borel reducibility hierarchy of equivalence relations.

Slides available here.

**Rod Downey, University of Wellington**

*Title*: Algorithmically Random Trigonometric Series

*Abstract*: Recently, we have seen the uses of the theory of algorithmic randomness to solve questions in classical mathematics. Some of these are purely classical and some have a more algorithmic feel. We will discuss some of these initiatives, illustrating the ideas via some longstanding questions in the theory of random trigonometric series. In particular, Rademacher [Rad22], Steinhaus [Ste30] and Paley and Zygmund [PZ30a, PZ30b, PZ32]initiated the extensive study of random series. Using the theory of algorithmic randomness, which is a mix of computability theory and probability theory, we investigate the effective content of some classical theorems.

We discuss how this is related to an old question of Kahane and Bollobas [Bol01], as reported in [DGTta]. We also discuss how considerations of such algorithmic questions about random series seems to lead to new notions of algorithmic randomness.

[Bol01] Bela Bollobas. Random graphs, volume 73 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second

edition, 2001.

[DGTta] R. Downey, N. Greenberg and A. Tanggarra. Algorithmically random series, and uses of algorithmic randomness in mathematics. Submitted.

[PZ30a] R. E. A. C. Paley and A. Zygmund. On some series of functions (1). Mathematical Proceedings of the Cambridge Philosophical Society, 26(4):337–257, 1930.

[PZ30b] R. E. A. C. Paley and A. Zygmund. On some series of functions (2). Mathematical Proceedings of the Cambridge Philosophical Society,26(4):458–474, 1930.

[PZ32] R. E. A. C. Paley and A. Zygmund. On some series of functions, (3). Mathematical Proceedings of the Cambridge Philosophical Society, 28(2):190–205, 1932.

[Rad22] H. Rademacher. Einige sätze über Reihen von allgemeinen Orthogonal-Funktionen. Mathematische Annalen, 87:112–138, 1922.

[Ste30] Hugo Steinhaus. Uber die wahrscheinlichkeit dafur, das der konvergenzkreis einer Potenzreihe ihre natürliche Grenze ist. Mathematische Zeitschrift, 31(1):408–416, 1930.

Slides available here.

**Kentaro Fujimoto & Philipp Schlicht, University of Bristol**

*Title*: Some open problems in second-order set theory

Kentaro Fujimoto’s slides are available here.

Philipp Schlicht’s slides are available here.

**Richard Matthews, University of Creteil, Paris**

*Title*: A guide to Krivine Realizability

*Abstract*: The method of realizability was first developed by Kleene and is seen as a way to extract computational content from mathematical proofs. Traditionally, these models only satisfy intuitionistic logic, however the method was extended by Krivine to produce models which satisfy full classical logic and even Zermelo Fraenkel set theory with choice. In this talk we will discuss how to construct realizability models of ZF and its connections with intuitionistic realizability, double negation translations and the method of forcing. We will then present recent results concerning ordinals and large cardinals in these realizability models. This is joint work with Laura Fontanella and Guillaume Geoffroy.

Slides available here.

**Fedor Pakhomov, University of Ghent**

*Title*: On limits of incompleteness theorems

*Abstract*: In this talk I will give a survey of several recent results

about the limits of incompletess theorems. Based on the papers:

[1] Pakhomov, F., & Visser, A. (2022). Finitely axiomatized theories

lack self‐comprehension. Bulletin of the London Mathematical Society, 54(6), 2513-2531.

[2] Murwanashyaka, J., Pakhomov, F., & Visser, A. (2023). There are no

minimal essentially undecidable theories. Journal of Logic and Computation.

Slides available here.

**Paul Shafer, University of Leeds**

*Title*: The logical and computational strength of inside/outside Ramsey theorems

*Abstract*: Rival and Sands proved that every infinite graph G contains an infinite subset H such that every vertex of G is adjacent to precisely none, one, or infinitely many vertices of H. We call this result an inside/outside Ramsey theorem because the conclusion provides information about vertices that are inside of H and about vertices that are outside of H. Rival and Sands also proved a similar statement for infinite partial orders of finite width. We analyze the strength of these theorems from the perspective of reverse mathematics and the Weihrauch degrees. We find that they give the first examples from the modern general mathematics literature of theorems that are equivalent to the double jump of weak König’s lemma in the Weihrauch degrees and of theorems that are equivalent to the ascending/descending sequence principle (plus Sigma_2 induction in some cases) in reverse mathematics. This work is joint with Marta Fiori Carones, Alberto Marcone, and Giovanni Soldà.

Slides available here.

**Johannes Stern, University of Bristol**

*Title*: From Intuitionistic Kripke Frames to Strong Kleene Supervaluation and Theories of Naive Truth.

*Abstract*: I show how starting from intuionistic Kripke frames one can develop a supervaluational framework that lends itself to inductively defining a truth predicate in the presence of an intutionistic conditional.

Slides available here.

**Xinhe Wu, University of Bristol**

*Title*: Full and Mixed Models

*Abstract*: In this talk, I discuss two special kinds of Boolean-valued models: full models and mixed models. I show that these models are more “classical” than the others, as some classical model-theoretic results can only be generalized to these them. In particular, the Łoś ultraproduct theorem and (a strong version of) downward Lowenheim-Skolem theorem can only be generalized to full models, and the theorem that every countably incomplete ultraproduct is ω1-saturated and the theorem that Σ^1_1 formulas are preserved under ultraproducts can only be generalized to mixed models.

Slides available here.

**Bokai Yao, University of Notre Dame**

*Title*: Reflection with Absolute Generality

*Abstract*: Traditionally, reflection principles in set theory claim that the set-theoretic universe is indescribable. It is natural to consider reflection principles with absolute generality, which asserts that the universe containing everything, including sets and urelements, is indescribable. In the first part of this talk, I will consider the first-order reflection principle in urelement set theory. With the Axiom of Choice, first-order reflection holds just in case urelements are arranged in a certain way, and this equivalence falls apart without AC. In the second part of this talk, I will present my joint work with Joel Hamkins on second-order reflection principles with urelements. A standard version of second-order reflection, due to Paul Bernays, is often considered as a weak large cardinal axiom in set theory. With abundant urelements, however, Bernays’ second-order reflection principle interprets a supercompact cardinal.

Slides available here.

**Schedule:**

**Friday 30th June**

14:00-15:00 – Richard Matthews

15:00-15:30 – Break

15:30-16:30 – Rod Downey

16:30-16:45 – Break

16:45-17:15 – Xinhe Wu

17:15-17:45 – Kentaro Fujimoto & Philipp Schlicht

**Saturday 1st July**

09:00-10:00 – Fedor Pakhomov

10:00-10:30 – Break

10:30-11:30 – Paul Shafer

11:30-12:00 – Sam Coskey

12:00-14:00 – Lunch

14:00-14:30 – Bokai Yao

14:30-15:30 – Johannes Stern

All talks will take place in **G.13, Fry Building**.

**List of participants:**

Sam Coskey – University College London

Joseph Deakin – University of Cambridge

Rod Downey – University of Wellington

Ugur Efem – Dyson Institute of Engineering and Technology

Kentaro Fujimoto – University of Bristol

Colin Harling – NSSL

Charles Harris – University of Bristol

Alex Kavvos – University of Bristol

Clara List – Universität Hamburg

Xianrui Liu – University of Bristol

Richard Matthews – University of Creteil, Paris

Michael Mooney – University of Bristol

Fedor Pakhomov – University of Ghent

Sherwin Pereira – University of Bristol

Simone Picenni – University of Bristol

Cécilia Pradic – Swansea University

Paul Shafer – University of Leeds

Philipp Schlicht – University of Bristol

Johannes Stern – University of Bristol

Esme Weil – University of Bristol

Xinhe Wu – University of Bristol

Bokai Yao – University of Notre Dame

**Registration**

This event has now passed and registration is closed.

This event is organised by Kentaro Fujimoto (kentaro.fujimoto@bristol.ac.uk) and Philipp Schlicht (philipp.schlicht@bristol.ac.uk).

For practical information please contact maths-conference-administrator@bristol.ac.uk.

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