### The Leeds Logic group

Mathematical logic, a young subject, has developed over the last 30 years into an amalgam of fast-moving disciplines, each with its own sophisticated techniques. These are linked by profound common concerns, around definability, decidability, effectiveness and computability, the nature of the continuum, and foundations. Some branches are highly multidisciplinary and force the researcher to be fully conversant with other fields (e.g. algebra, computer science).

The Logic Group at the University of Leeds is one of the largest and most active worldwide, with a long uninterrupted tradition dating back to 1951, when its founder Martin Löb moved to Leeds. It has an international reputation for research in most of the main areas of mathematical logic – computability theory, model theory, set theory, proof theory, and in applications to algebra, analysis, number theory and theoretical computer science.

### Activities

• Upcoming & past events.
• The Löb lectures.
• The Models and Sets Seminar.
• Soon to appear, future announcements and archives of: Leeds-Ghent Virtual Logic Seminar; Logic Seminar; Algebra, Logic and Algorithms Seminar; Model Theory Seminar (replaced by Models and Sets); Proofs, Constructions and Computations (replaced by Proofs, Constructions, Computations, Categories).

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Abstract

It is well-known that forcings preserve $\mathsf{ZFC}$, i.e., any set generic extension of any model of $\mathsf{ZFC}$ is again a model of $\mathsf{ZFC}$. How about the Axiom of Determinacy ($\mathsf{AD}$) under $\mathsf{ZF}$? It is not difficult to see that Cohen forcing always destroys $\mathsf{AD}$, i.e., any set generic extension of a model of $\mathsf{ZF}+ \mathsf{AD}$ via Cohen forcing is not a model of $\mathsf{AD}$. Actually it is open whether there is a forcing which adds a new real while preserving $\mathsf{AD}$. In this talk, we present some results on preservation & non-preservation of $\mathsf{AD}$ via forcings, whose details are as follows:

1. Starting with a model of $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R))$, any forcing increasing $\Theta$ destroys $\mathsf{AD}$.
2. It is consistent relative to $\mathsf{ZF} + \mathsf{AD}_R$ that $\mathsf{ZF} + \mathsf{AD}^{+} +$ There is a forcing which increases $\Theta$ while preserving $\mathsf{AD}$.
3. In $\mathsf{ZF}$, no forcings on the reals preserve $\mathsf{AD}$. (This is an improvement of the result of Chan and Jackson where they additionally assumed $\Theta$ is regular.)
4. In $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R)) + \Theta$ is regular, there is a forcing on $\Theta$ which adds a new subset of $\Theta$ while preserving $\mathsf{AD}$.

This is joint work with Nam Trang.

more

Abstract

An integral domain is Prüfer if its localisation at each maximal ideal is a valuation domain. Many classically important rings are Prüfer domains. For instance, they include Dedekind domains and hence rings of integers of number fields; Bézout domains and hence the ring of complex entire functions and the ring of algebraic integers; the ring of integer valued polynomials with rational coefficients and the real holomorphy rings of formally real fields.

Over the last 15 years, efforts have been made to characterise when the theory of modules of (particular types of) Prüfer domains are decidable. I will give an overview of such decidability results culminating in recently obtained elementary conditions completely characterising when the theory of modules of an arbitrary Prüfer domain is decidable.

more

Abstract

In pursuit of an understanding of the relations between compactness and approximation principles we address the question: To what extent do compactness principles assert the existence of a diamond sequence? It is well known that a cardinal $\kappa$ that satisfies a sufficiently strong compactness assumption must also carry a diamond sequence. However, other results have shown that certain weak large cardinal assumptions are consistent with the failure of the full diamond principle. We will discuss this gap and describe recent results with Jing Zhang which connect this problem to the existence of a certain global notion of cardinal arithmetic scales.

more

Abstract

We sketch a proof of mutual interpretability of Robinson arithmetic and a weak finitely axiomatized theory of concatenation.

more

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Abstract

It is well-known that forcings preserve $\mathsf{ZFC}$, i.e., any set generic extension of any model of $\mathsf{ZFC}$ is again a model of $\mathsf{ZFC}$. How about the Axiom of Determinacy ($\mathsf{AD}$) under $\mathsf{ZF}$? It is not difficult to see that Cohen forcing always destroys $\mathsf{AD}$, i.e., any set generic extension of a model of $\mathsf{ZF}+ \mathsf{AD}$ via Cohen forcing is not a model of $\mathsf{AD}$. Actually it is open whether there is a forcing which adds a new real while preserving $\mathsf{AD}$. In this talk, we present some results on preservation & non-preservation of $\mathsf{AD}$ via forcings, whose details are as follows:

1. Starting with a model of $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R))$, any forcing increasing $\Theta$ destroys $\mathsf{AD}$.
2. It is consistent relative to $\mathsf{ZF} + \mathsf{AD}_R$ that $\mathsf{ZF} + \mathsf{AD}^{+} +$ There is a forcing which increases $\Theta$ while preserving $\mathsf{AD}$.
3. In $\mathsf{ZF}$, no forcings on the reals preserve $\mathsf{AD}$. (This is an improvement of the result of Chan and Jackson where they additionally assumed $\Theta$ is regular.)
4. In $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R)) + \Theta$ is regular, there is a forcing on $\Theta$ which adds a new subset of $\Theta$ while preserving $\mathsf{AD}$.

This is joint work with Nam Trang.

more

Abstract

An integral domain is Prüfer if its localisation at each maximal ideal is a valuation domain. Many classically important rings are Prüfer domains. For instance, they include Dedekind domains and hence rings of integers of number fields; Bézout domains and hence the ring of complex entire functions and the ring of algebraic integers; the ring of integer valued polynomials with rational coefficients and the real holomorphy rings of formally real fields.

Over the last 15 years, efforts have been made to characterise when the theory of modules of (particular types of) Prüfer domains are decidable. I will give an overview of such decidability results culminating in recently obtained elementary conditions completely characterising when the theory of modules of an arbitrary Prüfer domain is decidable.

more

Abstract

In pursuit of an understanding of the relations between compactness and approximation principles we address the question: To what extent do compactness principles assert the existence of a diamond sequence? It is well known that a cardinal $\kappa$ that satisfies a sufficiently strong compactness assumption must also carry a diamond sequence. However, other results have shown that certain weak large cardinal assumptions are consistent with the failure of the full diamond principle. We will discuss this gap and describe recent results with Jing Zhang which connect this problem to the existence of a certain global notion of cardinal arithmetic scales.

more

Abstract

We sketch a proof of mutual interpretability of Robinson arithmetic and a weak finitely axiomatized theory of concatenation.

more
• Dr Sebastian Eterović
• Dr Zeinab Galal
• Dr Federico Olimpieri
• Dr Jonathan Schilhan
• Dr Jiachen Yuan