Archives of: Michigan Logic Seminar

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.

David J. Fernández Bretón: mathfrak p=mathfrak t, III

Tuesday, April 18, 2017, from 4 to 5:30pm
East Hall, room 3096

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

Title: mathfrak p=mathfrak t, III

Abstract:

This is the third and last talk in the series (reasonably self-contained for those who missed any number of previous parts). I will continue to present the proof, due to Maryanthe Malliaris and Saharon Shelah in 2012, that the cardinal invariants p and t are equal, which constitutes an extremely important result in the theory of Cardinal Characteristics of the Continuum.

David J. Fernández Bretón: mathfrak p=mathfrak t, II

Thursday, April 6, 2017, from 4 to 5:30pm
East Hall, room 3088

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

Title: mathfrak p=mathfrak t, II

Abstract:

This is the second in a series of (hopefully at most) three talks, and it will be reasonably self-contained for those who missed the first part. I will continue to present the proof, due to Maryanthe Malliaris and Saharon Shelah in 2012, that the cardinal invariants p and t are equal, which constitutes an extremely important result in the theory of Cardinal Characteristics of the Continuum.

David J. Fernández Bretón: mathfrak p=mathfrak t

Thursday, March 30, 2017, from 4 to 5:30pm
East Hall, room 3088

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

Title: mathfrak p=mathfrak t

Abstract:

In a series of (hopefully at most) two talks, I will present the proof, due to Maryanthe Malliaris and Saharon Shelah in 2012, that the cardinal invariants p and t are equal, which constitutes an extremely important result in the theory of Cardinal Characteristics of the Continuum.

Andres Caicedo: MRP and squares, II

Thursday, March 23, 2017, from 4 to 5:30pm
East Hall, room 3088

Speaker: Andres Caicedo (Math Reviews)

Title: MRP and squares, II

Abstract:

Justin Moore’s mapping reflection principle (MRP) seems to capture the consistency strength of PFA, since it implies the failure of square. I continue the presentation of some refinements and extensions of this result. They are due to a variety of authors, and some remain unpublished.

Andres Caicedo: MRP and squares

Thursday, March 16, 2017, from 4 to 5:30pm
East Hall, room 2866

Speaker: Andres Caicedo (Math Reviews)

Title: MRP and squares

Abstract:

Justin Moore’s mapping reflection principle (MRP) seems to capture the consistency strength of PFA, since it implies the failure of square. I present some refinements and extensions of this result. They are due to a variety of authors, and some remain unpublished.

Ioannis Souldatos: L_{omega_1,omega}-sentences with maximal models in two cardinalities, part II

Thursday, February 16, 2017, from 4 to 5:30pm
East Hall, room 2866

Speaker: Ioannis Souldatos (University of Detroit Mercy)

Title: L_{omega_1,omega}-sentences with maximal models in two cardinalities, part II

Abstract:

This will be part II of the talk on complete L_{omega_1,omega}-sentences with maximal models
in (at least) two cardinalities. The talk will be self-contained.

Sample theorems

Theorem: If kappa is homogeneously characterizable and mu is the least such that 2^mu>=kappa, then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities
2^lambda, for all mu<=lambdaaleph_0 is the least such that mu^omega>=kappa, then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities kappa^omega and kappa.

Theorem (Baldwin-Shelah) If mu is the first measurable cardinal and phi belongs to L_{omega_1,omega}, then no model of phi of size greater or equal to mu is maximal with respect to the L_{omega_1,omega}-elementary substructure relation.

Ioannis Souldatos: L_{omega_1,omega}-sentences with maximal models in two cardinalities

Thursday, February 9, 2017, from 4 to 5:30pm
East Hall, room 2866

Speaker: Ioannis Souldatos (University of Detroit Mercy)

Title: L_{omega_1,omega}-sentences with maximal models in two cardinalities

Abstract:

In this talk, we will present some examples on complete L_{omega_1,omega}-sentences with maximal models in (at least) two cardinalities.

Sample theorems:

Theorem: There is a complete L_{omega_1,omega}-sentence that characterizes aleph_2 and has maximal models in aleph_1 and aleph_2.

Theorem: Assume 2^{aleph_0}>aleph_n. Then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities 2^{aleph_0}, 2^{aleph_1},…,2^{aleph_n}.

The main construction behind these theorems is a refinement of a construction of J. Knight. This is recent work of J. Baldwin and the speaker.

Ioannis Souldatos: A survey on the effect of set-theory on models of L_{omega_1,omega}-sentences

Thursday, November 17, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: Ioannis Souldatos (University of Detroit Mercy)

Title: A survey on the effect of set-theory on models of L_{omega_1,omega}-sentences

Abstract:

The model-existence spectrum of an L_{omega_1,omega}-sentence phi is the set of all cardinals on which phi has a model.
During the talk we will survey known theorems about the model-existence spectra of L_{omega_1,omega}-sentences, focusing on how the underlying set-theory affects these spectra.

Andres Caicedo: Preserving sequences of stationary subsets of omega_1

Thursday, November 10, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: Andres Caicedo (Math Reviews)

Title: Preserving sequences of stationary subsets of omega_1

Abstract:

Let M be an inner model that computes omega_1 correctly. We show two results (due to Stevo Todorcevic and Paul Larson) on whether there is in M a partition of omega_1 into infinitely many sets that are stationary from the point of view of V.