Archives of: Michigan Logic Seminar

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 They are due to a variety of authors, and some remain unpublished.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.

David Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, IV

Thursday, November 3, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: David Fernández Bretón (University of Michigan)

Title: Strong failures of higher analogs of Hindman’s theorem, IV

Abstract:

This is talk 4 out of 4. Last time we introduced a set-theoretic principle S(kappa,theta), which implies a particular anti-Ramsey theoretic result on all groups of cardinality kappa. In this talk we will prove that S(omega_1,omega_1) holds, and also that S(kappa,omega) holds whenever kappa has uncountable cofinality equals the cofinality of the powerset of a cardinal lambda such that lambda^{<lambda}=lambda (this last result has a nice implication for the additive group of real numbers).