Archives of: Michigan Logic Seminar

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 They are due to a variety of authors, and some remain unpublished.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 They are due to a variety of authors, and some remain unpublished.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

David J. Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, III

Thursday, October 27, 2016, from 4 to 5:30pm
East Hall, room 3096

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

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

Abstract:

Assuming the Continuum Hypothesis, Hindman, Leader and Strauss recently exhibited a colouring of the real line with two colours such that, for every uncountable set of reals, the collection of pairwise sums of these reals is panchromatic. We will show a few generalizations of these results, obtaining colourings both of the real line, and of other abelian groups, in many colours, satisfying similar anti-Ramsey-theoretic properties. This is talk number 3 out of n (where n is still TBD), and its contents are joint work with Assaf Rinot.

David J. Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, II

Thursday, October 20, 2016, from 4 to 5:30pm
East Hall, room 3096

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

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

Abstract:

(One of the versions of) Hindman’s theorem states that, whenever we partition an infinite abelian group G in two cells, there exists an infinite subset X of G such that the set FS(X) consisting of all sums of finitely many distinct elements of X is entirely contained within one of the cells of the partition. In this talk we will show that, when one attempts to replace both instances of “infinite” with “uncountable” in the theorem above, the resulting statement is not only false, but actually very false. This is talk 2 out of n (where n is a still unknown countable ordinal greater than or equal to 2). Joint work with Assaf Rinot.

David J. Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, I

Thursday, October 13, 2016, from 4 to 5:30pm
East Hall, room 3096

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

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

Abstract:

(One of the versions of) Hindman’s theorem states that, whenever we partition an infinite abelian group G in two cells, there exists an infinite subset X of G such that the set FS(X) consisting of all sums of finitely many distinct elements of X is entirely contained within one of the cells of the partition. In this talk we will show that, when one attempts to replace both instances of “infinite” with “uncountable” in the theorem above, the resulting statement is not only false, but actually very false. This is talk 1 out of n (where n is a still unknown nonzero countable ordinal). Joint work with Assaf Rinot.