Swiss Logic Gathering 2025

The Swiss Logic Gathering 2025 will take place on December 5 at ETH Zürich in the main building, at Floor F room HG F 33.1. The address of the building is Rämistrasse 101, 8092 Zürich, Schweiz.

Schedule

Abstracts

Originally studied in the context of valued fields, ranks function as order-theoretic invariants for linearly ordered structures of varying algebraic complexity. Most generally, the rank of an ordered algebraic structure is the order type of a certain collection of convex substructures, ordered by set inclusion: For an ordered field, this collection consists of all of proper convex subrings (i.e. the non-trivial convex valuation rings of that field); for ordered abelian groups, it consists of all proper convex subgroups; for ordered sets (without any further algebraic structure), the collection contains all proper final segments.

Studying the ranks of ordered fields, ordered abelian groups and ordered sets becomes of special interest in the presence of valuations. More specifically, the finest convex valuation on an ordered field – its so-called natural valuation – connects a unique ordered abelian group to that field. Likewise, each ordered abelian group has a natural valuation resulting in an ordered value set. Now the rank of any ordered field coincides with the rank of its value group under the natural valuation; likewise the rank of any ordered abelian group is identical to the rank of its value set under the natural valuation. Thus, for an ordered field, all three ranks coincide: at the field, the value group and the value set levels.

From a model-theoretic perspective, ranks can be limited to collections of convex substructures that are first-order definable, introducing the concept of definable ranks. For instance, the definable rank of an ordered field is the order type of the set of its first-order definable convex valuation rings. While it is readily established that definable ranks generally do not coincide at the field, value group, and value set levels, the question remains as to what order-theoretic relationships still exist between them.

In my talk, I will outline the classical correspondence between the ranks on field, group and set level. I will then present some preliminary results from joint work with Salma Kuhlmann and Lasse Vogel [Definable Ranks, Preprint, 2025, arXiv:2506.00443]. All relevant notions from valuation theory will be introduced.

In the recent years there has been an increasing interest in the Proof Theory community on non-wellfounded proofs, i.e., proofs whose chain of reasoning may go backwards without and end. Thanks to the work of Savateev and Shamkanov this kind of proofs were used in the proof theoretical study of provability logics GL and Grz. In joint work with other authors, the use of non-wellfounded proofs for studying provability logics has been simplified and extended. One of the benefits of non-wellfounded proofs in the context of provability logic is that, unlike the usual wellfounded calculi, has good propoerties with respect to the polarities of formulas. In this talk, apart of a brief introduction to the previous panorama, we will present how to exploit this feature to obtain Uniform Lyndon interpolation of GL via non-wellfounded proofs. This methodology can be used for other provability logics, although its full extension is still unclear.

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. This is joint work with Shelah.

Arguably, the three most well-known types of ultrafilters over the natural numbers are the Ramsey ultrafilters, the P-points and the Q-points. Under set-theoretic assumptions such as the Continuum Hypothesis or Martin’s Axiom for countable partial orders, these ultrafilters are abundant, in the sense that there exist 2^(2^ω) of them - as many as there are ultrafilters over ω. However, the existence of each of these types cannot be proven in ZFC alone, as was shown by Kunen (1975), Shelah (1977) and Miller (1980), respectively.

We will talk about the consistency with ZFC of the statement ``There exist exactly n Q-points up to isomorphism”, for any natural number n. This result extends a recent corollary of Mildenberger (2024), who showed the consistency of the above statement for n=1,2. It also complements an older result of Shelah (1998), who showed the consistency of the above statement for Ramsey ultrafilters and P-points in place of Q-points.

This is joint work with Lorenz Halbeisen and Tan Özalp.

The relationships between various forms of interpolation for propositional logics and amalgamation for classes of algebraic structures have received considerable attention in the literature in the frameworks of model theory, abstract algebraic logic, universal algebra, and residuated structures. A well-known theorem states that a variety with the congruence extension property (CEP) has the amalgamation property (AP) if and only if it has the deductive interpolation property (DIP). This “bridge theorem” provides a powerful technique for establishing the deductive interpolation property via the amalgamation property, and vice versa. However, for varieties that lack the congruence extension property, failure of the amalgamation property does not necessarily imply failure of the deductive interpolation property. To provide a route to developing a more systematic understanding of the DIP, we will  explore the gap between the DIP and the AP, and review the solutions found in the literature to the problem of describing an algebraic property that corresponds directly to the DIP. Finally, we will address some decidability questions concerning these properties.  This is joint work with Wesley Fussner, George Metcalfe and Simon Santschi.