Archives of: Michigan Logic Seminar

David J. Fernández Bretón: Models of set theory with union ultrafilters and small covering of meagre

Thursday, February 15, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Models of set theory with union ultrafilters and small covering of meagre

Abstract:

Union ultrafilters are ultrafilters that arise naturally from Hindman’s finite unions theorem, in much the same way that selective ultrafilters arise from Ramsey’s theorem, and they are very important objects from the perspective of algebra in the Cech–Stone compactification. The existence of union ultrafilters is known to be independent from the ZFC axioms (due to Hindman and Blass–Hindman), and is known to follow from a number of set-theoretic hypothesis, of which the weakest one is that the covering of meagre equals the continuum (this is due to Eisworth). I will show that such hypothesis is not a necessary condition, by exhibiting a number of different models of ZFC that have a covering of meagre strictly less than the continuum, while at the same time satisfying the existence of union ultrafilters.

David J. Fernández Bretón: More Ramsey-theoretic statements: uncountably many colours, finite monochromatic sets

Thursday, December 7, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: More Ramsey-theoretic statements: uncountably many colours, finite monochromatic sets

Abstract:

Hindman’s theorem states that for every colouring of an infinite abelian group with finitely many colours, there will be an infinite set whose finite sums are monochromatic. Increasing the number of colours to infinitely many makes the theorem fail, as does keeping the number of colours finite but requiring an uncountable monochromatic finite-sum set. Recently Komjáth discovered, however, that by increasing the number of colours to be infinite (even uncountable), and at the same time decreasing the size of our desired monochromatic set to a finite number, it is possible to obtain some positive Ramsey-theoretic results. In this talk I will discuss some of these results, as well as some improvements and generalizations of them that I found over the summer, jointly with my REU student Sung-Hyup Lee.

Andres Caicedo: Real-valued measurability and the extent of Lebesgue measure (II)

Thursday, November 30, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Andres Caicedo (Math Reviews)

Title: Real-valued measurability and the extent of Lebesgue measure (II)

Abstract:

On this second talk I begin with Solovay’s characterization of real-valued measurability in terms of generic elementary embeddings, and build on results of Judah to prove that if there is an atomlessly measurable cardinal, then all (boldface) Delta-1-3 sets of reals are Lebesgue measurable. This is optimal in two respects: Just from the existence of measurable cardinals we cannot prove that lightface Delta-1-3 sets are measurable, and there are models with atomlessly measurable cardinals where there is a non-measurable Sigma-1-3 set. I will also discuss some related results.

Andres Caicedo: Real-valued measurability and the extent of Lebesgue measure

Thursday, November 9, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Andres Caicedo (Math Reviews)

Title: Real-valued measurability and the extent of Lebesgue measure

Abstract:

The existence of an atomlessly measurable cardinal is equivalent to the existence of a measure extending Lebesgue measure and defined on all sets of reals. I’ll start the talk with some background on real-valued measurability, and proceed to argue that the assumption that there is some such cardinal actually has an effect on the extent of Lebesgue measure itself. The result goes beyond what can be granted arguing merely in terms of consistency strength.

Danny Nguyen: Presburger Arithmetic and its computational complexity

Thursday, November 2, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Danny Nguyen (University of California, Los Angeles)

Title: Presburger Arithmetic and its computational complexity

Abstract:

Presburger Arithmetic (PA) is a classical topic in logic, with numerous connections to computer science and combinatorics. Formally, is the first order structure on the integers with only additions and inequalities. Despite its long history, many problems in PA have remained unsolved until recently. We study the complexity of decision problems in PA, and classify them according to hierarchy levels. Along the way, connections to Integer Programming and Optimization will be explained. The talk will be self contained and assumes no prior knowledge of the subject. Joint work with Igor Pak.

David J. Fernández Bretón: Higher degree versions of the Central Sets Theorem, II

Thursday, October 26, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Higher degree versions of the Central Sets Theorem, II

Abstract:

The Central Sets Theorem is a Ramsey-theoretic result due to Furstenberg, from 1981, and multiple generalizations of it (in a variety of different directions) have been proved afterwards (to the best of my knowledge, the currently most general statement is due to De, Hindman and Strauss in 2008, but there are also many relevant results due to Bergelson). This is the second of a series of two talks, where we will explain how to interpret the Central Sets Theorem as a statement about linear polynomials in a polynomial ring with countably many variables, and prove a couple of natural generalizations involving polynomials of higher degree. The main tool that we use in our proof is the algebra of the Cech–Stone compactification (that is, these are “ultrafilter proofs”).

David J. Fernández Bretón: Higher degree versions of the Central Sets Theorem

Thursday, October 12, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Higher degree versions of the Central Sets Theorem

Abstract:

The Central Sets Theorem is a Ramsey-theoretic result due to Furstenberg, from 1981, and multiple generalizations of it (in a variety of different directions) have been proved afterwards (to the best of my knowledge, the currently most general statement is due to De, Hindman and Strauss in 2008, but there are also many relevant results due to Bergelson). In this series of two talks, we will explain how to interpret the Central Sets Theorem as a statement about linear polynomials in a polynomial ring with countably many variables, and prove a couple of natural generalizations involving polynomials of higher degree. In order to make this exposition self-contained, we will spend most of the first talk providing an overview of the techniques from algebra in the Cech–Stone compactification, which is the main tool that we use in our proof.

Abhijit Dasgupta: Axioms for complete elementary extensions

Thursday, October 5, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Abhijit Dasgupta (University of Detroit Mercy)

Title: Axioms for complete elementary extensions

Abstract:

We give an axiomatic framework for “logicless non-standard analysis”, using the notion of partial functions as a primitive.

Simon Cho: A Category Theoretic Perspective on Continuous Logic, II

Thursday, September 28, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Simon Cho (University of Michigan)

Title: A Category Theoretic Perspective on Continuous Logic, II

Abstract:

Although classical model theory is largely formulated in terms of the framework of sets, there is a rich theory that casts model theoretic structures in a category theoretic setting, a project which began with Lawvere’s thesis on “functorial semantics of algebraic theories” and has since grown into an important subfield of category theory. This interface between classical model theory and category theory continues to be an active area of research today.

In parallel, Lawvere also showed that structures – such as metric spaces – seemingly unrelated to categories arose naturally as examples of categories with appropriate enrichments V (for example V=R in the case of metric spaces). Now continuous logic/metric model theory is a generalization of classical model theory that, roughly, replaces sets with metric spaces and equality with the metric; a natural question to ask is whether the above perspective on metric spaces combines with the way of interpreting classical logic into category theory to produce a way to interpret continuous logic into enriched category theory. This talk will answer this in the affirmative, under reasonable conditions.

Simon Cho: A Category Theoretic Perspective on Continuous Logic

Thursday, September 21, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Simon Cho (University of Michigan)

Title: A Category Theoretic Perspective on Continuous Logic

Abstract:

Although classical model theory is largely formulated in terms of the framework of sets, there is a rich theory that casts model theoretic structures in a category theoretic setting, a project which began with Lawvere’s thesis on “functorial semantics of algebraic theories” and has since grown into an important subfield of category theory. This interface between classical model theory and category theory continues to be an active area of research today.

In parallel, Lawvere also showed that structures – such as metric spaces – seemingly unrelated to categories arose naturally as examples of categories with appropriate enrichments V (for example V=R in the case of metric spaces). Now continuous logic/metric model theory is a generalization of classical model theory that, roughly, replaces sets with metric spaces and equality with the metric; a natural question to ask is whether the above perspective on metric spaces combines with the way of interpreting classical logic into category theory to produce a way to interpret continuous logic into enriched category theory. This talk will answer this in the affirmative, under reasonable conditions. The talk will make every effort to be self-contained, and as such will assume little to no prior knowledge of category theory.