Location: MALL
Title: The limit of Gödel's first incompleteness theorem
NOTE date and time change: Fri 4pm
In this talk, we discuss the limit of the first incompleteness theorem (G1). It is well known G1 can be extended to both extensions and weak sub-systems of PA. We examine the question: are there minimal theories for which G1 holds. The answer of this question depends on how we define the notion of minimality. We discuss different answers of this question based on varied notions of minimality.
The notion of interpretation provides us a general method to compare different theories in distinct languages. We examine the question: are there minimal theories for which G1 holds with respect to interpretability. It is known that G1 holds for essentially undecidable theories, and there are no minimal essentially undecidable theories with respect to interpretability. G1 holds for effectively inseparable (EI) theories and the notion of effective inseparability is much stronger than essential undecidability. A natural question is: are there minimal EI theories with respect to interpretability? We negatively answer this question and prove that there are no minimal effectively inseparable theories with respect to interpretability: for any EI theory T, we can effectively find a theory which is EI and strictly weaker than T with respect to interpretability. Moreover, we prove that there are no minimal finitely axiomatizable EI theories with respect to interpretability.
Finally, we give a summary of the similarities and differences between logical incompleteness and mathematical incompleteness based on technical evidences and philosophical reflections.
Location: Roger Stevens LT23 (8.23)
Title: Effective Pila–Wilkie bounds for Pfaffian sets with some diophantine applications
NOTE location change: Roger Stevens LT23 (8.23)
Following critical insights of Pila and Zannier, there are by now many applications of model theory to diophantine geometry arising from the celebrated counting theorem of Pila and Wilkie and its variants. The original Pila–Wilkie Theorem bounds the number of rational points of bounded numerator and denominator lying on (the transcendental parts of) sets definable in o-minimal expansions of the real field. However, the proof of this theorem (and that of its variants) does not provide an effective bound, which limits the precision of its applications. I will discuss some joint work with Gal Binyamini, Gareth O. Jones and Harry Schmidt in which we obtained effective forms of the Pila–Wilkie Theorem and its variants for sets definable in various structures described by Pfaffian functions (including an effective Yomdin–Gromov parameterization result for sets defined using restricted Pfaffian functions), and then used these effective estimates to derive several effective diophantine applications, including an effective, uniform Manin–Mumford statement for products of elliptic curves with complex multiplication.
Location: Roger Stevens LT23 (8.23)
Title: Algebraic minimality of automorphism groups of countable homogeneous structures
NOTE location change: Roger Stevens LT23 (8.23)
Permutation groups of a countable set are Hausdorff topological groups with the pointwise convergence topology. A Hausdorff topological group G is minimal if every bijective continuous homomorphism from G to another Hausdorff topological group is a homeomorphism. The Zariski topology is defined in a natural way for any group. However, a permutation group with the Zariski topology is not necessarily a topological group. When the Zariski topology is a topological group then it is minimal. In this talk we investigate the Zariski topology for the automorphism groups of some countable homogeneous structures. This is a joint work with Javier de la Nuez González.
Location: Roger Stevens LT23 (8.23)
Title: O-minimality and finiteness of some atypical intersections for $Y(1)^n$
NOTE location change: Roger Stevens LT23 (8.23)
I will explain how o-minimality can be used to prove that, for a modular function f with Heegner divisor, there are only finitely many $n$-tuples of $f$-images of CM-points which are multiplicatively dependent. I will also discuss the relation between this result and the Zilber–Pink conjecture for atypical intersections.
Location: MALL
Title: Introducing a first-order theory of ordered transexponential fields
(joint work with Salma Kuhlmann)
Studying the growth properties of definable functions in o-minimal settings, Miller established the following remarkable growth dichotomy: an o-minimal expansion of an ordered field is either power bounded or admits a definable exponential function (see [2]). Going one step further in the hierarchy of growth, Miller’s dichotomy result naturally led to the question whether there exist o-minimal expansions of ordered fields that are not exponentially bounded. Recent research activity in this area is therefore motivated by the search for either an o-minimal expansion of an ordered exponential field by a transexponential function that eventually exceeds any iterate of the exponential or, contrarily, for a proof that any o-minimal expansion of an ordered field is already exponentially bounded.
In this talk, I will present our axiomatic approach towards the study of ordered fields equipped with a transexponential function from [1]. Namely, denoting by e an exponential and by T a compatible transexponential, we establish a first-order theory of ordered transexponential fields in which the functional equation $T(x + 1) = e(T(x))$ holds for any positive $x$. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural (i.e. the finest non-trivial convex) valuation. Moreover, I will illustrate construction methods for transexponentials on non-archimedean ordered exponential fields.
All relevant valuation theoretic background will be introduced.
[1] L. S. Krapp and S. Kuhlmann, ‘Ordered transexponential fields’, preprint, 2023, arXiv:2305.04607v1.
[2] C. Miller, ‘A Growth Dichotomy for O-minimal Expansions of Ordered Fields’, Logic: from Foundations to Applications (eds W. Hodges, M. Hyland, C. Steinhorn and J. Truss; Oxford Sci. Publ., Oxford Univ. Press, New York, 1996) 385–399.
Location: MALL
Title: Very large set axioms over Constructive Set Theories
One of the main areas of research in set theory is the study of large cardinal axioms and many of these can be characterised by the existence of elementary embeddings with certain properties. The guiding principle is then that the closer the domain and co-domain of the embedding is to the universe, the stronger the resulting large cardinal axiom. This leads naturally to the question of whether there is an elementary embedding of the universe into itself which is not the identity, and the least ordinal moved by such an embedding is known as a Reinhardt cardinal. While Kunen famously proved that no such embedding can exist if the universe satisfies ZFC, it is an open question in many subtheories of ZFC, most notably ZF (without Choice).
In this talk we will study elementary embeddings in the weaker context of intuitionistic set theories, that is set theories without the law of excluded middle. We shall observe that the ordinals can be very ill-behaved in this setting and therefore we will reformulate large cardinals by instead looking for large sets which capture the desired structural properties. We shall investigate the consistency strength of analogues to measurable cardinals, Reinhardt cardinals and many other similar ideas in terms of the standard ZFC large cardinal hierarchy.
Location: MALL
Title: A surjection from square onto power
Cantor proves that for any set $A$, there is no surjection from $A$ into its power set $\mathcal{P}(A)$. In this talk, we describe a construction of a ZF model. In this model, there is a set $A$ and a surjection from its square set $A^2$ onto its power set $\mathcal{P}(A)$. This indicates Cantor's Theorem is in some sense optimal. This is joint work with Guozhen Shen and Yinhe Peng.
Location: MALL
Title: Regular solutions of systems of transexponential-polynomial equations
It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function $E$ which satisfies $E(x+1) = \exp E(x)$. Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by $E$ could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.
An informal miniworkshop of early career model theorists in McMaster and (possibly formerly) Leeds. Carefully non-organised by Pantelis Eleftheriou and Vincenzo Mantova.
Location: MALL
Title: Hilbert polynomials for finitary matroids
Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets $A$ and $B$ of a commutative semigroup, the size of the sumset $A+tB$ is eventually polynomial in $t$. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero.
