The Toronto Set Theory Seminar is normally held on Fridays, from 1:30 to 3pm, in room 210 of the Fields Institute. For listing of talks from years earlier than 2006, see this page.

## Asger Tornquist: An invariant set-theoretic approach to Mathias’ theorem

Place: Fields Institute (Room 210)

Date: May 20, 2016 (13:30-15:00)

Speaker: Asger Tornquist

Title: An invariant set-theoretic approach to Mathias’ theorem

Abstract: Mathias proved in 1969 that there are no infinite analytic
maximal almost disjoint families of subsets of $\omega$. His proof is
essentially Ramsey theoretical. A few years ago I found a “classical” proof
of this theorem which uses a tree/derivative argument. But in this talk, I
will give yet another proof, this time one that is closer in spirit to
Mathias’ original proof, but which avoids the Ramsey theoretical machinery
by instead using a bit of garden-variety invariant descriptive set theory.

## Peter Koellner: Large Cardinals Beyond Choice

Place: Fields Institute (Room 210)

Date: May 13, 2016 (13:30-15:00)

Speaker: Peter Koellner

Title: Large Cardinals Beyond Choice

Abstract:

The hierarchy of large cardinals provides us with a canonical means to
climb the hierarchy of consistency strength. There have been many
purported inconsistency proofs of various large cardinal axioms. For
example, there have been many proofs purporting to show that
measurable cardinals are inconsistent. But to date the only proofs
that have stood the test of time are those which are rather
transparent and simple, the most notable example being Kunen’s proof
showing that Reinhardt cardinals are inconsistent. The Kunen result,
however, makes use of AC, and long standing open problem is whether
Reinhardt cardinals are consistent in the context of ZF.

In this talk I will survey the simple inconsistency proofs and then
raise the question of whether perhaps the large cardinal hierarchy
outstrips AC, passing through Reinhardt cardinals and reaching far
beyond. There are two main motivations for this investigation. First,
it is of interest in its own right to determine whether the hierarchy
of consistency strength outstrips AC. Perhaps there is an entire
“choiceless” large cardinal hierarchy, one which reaches new
consistency strengths and has fruitful applications. Second, since the
task of proving an inconsistency result becomes easier as one
strengthens the hypothesis, in the search for a deep inconsistency it
is reasonable to start with outlandishly strong large cardinal
assumptions and then work ones way down. This will lead to the
formulation of large cardinal axioms (in the context of ZF) that start
at the level of a Reinhardt cardinal and pass upward through Berkeley
cardinals (due to Woodin) and far beyond. Bagaria, Woodin, and I
have been charting out this new hierarchy. I will discuss what we have
found so far.

## Ari Brodsky: Custom-made Souslin trees

Place: Fields Institute (Room 210)

Date: May 2nd , 2016 (13:30-15:00)

Speaker: Ari Brodsky

Title: Custom-made Souslin trees

Abstract:  We propose a parametrized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $\kappa$-Souslin tree that applies also for $\kappa$ inaccessible.

Slides: AriBrodskyMay2016

## Juris Steprans: Graph embedding and the P-ideal dichotomy

Place: Fields Institute (Room 210)

Date: April 22nd , 2016 (13:30-15:00)

Speaker: Juris Steprans

Title: Graph embedding and the P-ideal dichotomy

Abstract: I will discuss a family of proofs of the consistency of a universal graph on $\omega_1$ with the failure of CH that rely on iterating reals and using the P-ideal dichotomy.

## Francisco Guevara: Analytic group topologies

Place: Fields Institute (Room 210)

Date: April 15th , 2016 (13:30-15:00)

Speaker: Francisco Guevera

Title: Analytic group topologies

Abstract: We study the effective version Malykhin’s question about the metrizability of (countable) Frechet groups and its natural generalization to metrizability of (countable) sequential groups of higher sequential order. A countable topological space $(X,\tau)$ is analytic if $\tau$ is analytic as a subset of the Cantor set $2^X$. By effective we mean the group topology is analytic. A space is sequential if all sequentially closed sets are closed. In sequential spaces, the sequential order is defined as the minimal ordinal $\alpha$ so that the closure of every set is obtained by applying the operation of adding limit points $\alpha$-many times. A sequential space has order $1$ iff it is Frechet. The results presented in the talk come from some works of A. Shibakov,  S. Todorcevic, and C. Uzcategui.

## Secil Tokgoz: Co-analytic sets of reals and Selection Principles

Place: Fields Institute (Room 210)

Date: April 8th, 2016 (13:30-15:00)

Speaker: Secil Tokgoz

Title: Co-analytic sets of reals  and Selection Principles

Abstract: In this talk we study one of the definable classes in the projective hierarchy, namely the family of co-analytic sets. A subset of a Polish space is called co-analytic if its complement is analytic. We will present some results on co-analytic sets of reals which have combinatorial properties of open covers.

## Antonio Aviles: Compact spaces of the first Baire class

Place: Fields Institute (Room 210)

Date: April 1st, 2016 (13:30-15:00)

Speaker: Antonio Aviles

Title: Compact spaces of the first Baire class

Abstract: A Rosenthal compactum is a pointwise compact set of functions of
the first Baire class on a Polish space. In a sense, the role of Rosenthal
compacta inside the larger class of general compacta is similar to the role
of analytic sets inside the class of general subsets of Polish spaces. Many
pathologies are discarded by their definable nature and structural theorems
appear instead. We will review some facts and we will present more recent
results in collaboration with S. Todorcevic and with A. Poveda and S.
Todorcevic.

## Jeffrey Bergfalk: Strong Homology and Set Theory

Place: Fields Institute (Stewart Library)

Date: March 28th, 2016 (15:30-16:30)

Speaker: Jeffrey Bergfalk

Title: Strong Homology and Set Theory

Abstract: We give some background to a number of independence results surrounding the question of the additivity of strong homology. These results center on the question of the vanishing of the higher derived limits of an inverse system $\mathbb{A}$ indexed by the functions from $\omega$ to $\omega$. Time permitting, we’ll show that $\text{lim}^1\mathbb{A}=0$ if and only if $\text{lim}^1\mathbb{A}_\kappa=0$, where $\mathbb{A}_\kappa$ is $\mathbb{A}$’s generalization to $\omega^\kappa$, with $\kappa>\omega$ arbitrary.

## Justin Moore: There may be no minimal non $\sigma$-scattered linear order

Place: Fields Institute (Stewart Library)

Date: March 28th, 2016 (14:00-15:00)

Speaker: Justin Moore

Title: There may be no minimal non $\sigma$-scattered linear order

Abstract:  In this talk we demonstrate that it is consistent that there is no linear order which is minimal with respect to being non $\sigma$-scattered.This shows a theorem of Laver, which asserts that the $\sigma$-scattered linear orders are well quasi-ordered is sharp. If time permits we will also prove that $PFA^+$ implies that every non $\sigma$-scattered linear order either contains a real type, an Aronszajn type, or a ladder system indexed by a stationary set, equipped with either the lexicographic of reverse lexicographic order. This is joint work with Hossein Lamei Ramandi.

## Iian Smythe: Towards a selective Gowers dichotomy

Place: Fields Institute (Stewart Library)

Date: March 28th, 2016 (13:00-14:00)

Speaker: Iian Smythe

Title: Towards a selective Gowers dichotomy

Abstract: Gowers’ famous dichotomy is an approximate Ramsey theorem for analytic partitions of the space of infinite block sequences in a Banach space, and has been used it to establish important classification results in Banach space theory. In work currently in progress, we attempt to isolate the combinatorial properties of the space of block sequences which enable these constructions, and prove that they can be carried out within  “selective” subfamilies. Under large cardinal assumptions, we extend these results to all definable partitions, with the goal of giving “complete combinatorics” for generic ultrafilters of block sequences.