# Set Theory Seminar at Fields Institute World Logic Day 2022 Talks

January 14th is World Logic Day (as designated by the UNESCO).

On this day the Toronto Set Theory Seminar at Fields Institute will hold a special session with 4 short talks (20 minutes each), showcasing recent research by faculty and postdocs at University of Toronto.

Here is a list of further world logic day events around the world.

## Program

### 13:30

####
Ivan Ongay-Valverde and Franklin D. Tall^{﹡}:
A New Topological Generalization of Descriptive Set Theory

We generalize the K-analytic spaces to the K-σ-projective spaces. We get an application to Selection Principles:

**Theorem.** The Axiom of σ-Projective Determinacy implies every
K-σ-projective Menger space is Hurewicz.

### 13:50

####
Ivan Ongay Valverde^{﹡} and Franklin D. Tall:
Upper semi-continuous compact-valued functions and the K-sigma-projective hierarchy

Completing the previous talk, we introduce USCCV functions (actually, multifunctions), which were employed in the study of K-analytic spaces, and show how to use them to prove the crucial:
**Theorem.** K-σ-projective spaces are projectively σ-projective.

### 14:10

####
Christopher J. Eagle, Clovis Hamel^{﹡}, Sandra MÃ¼ller, and Franklin D. Tall: An undecidable extension of Morley's theorem on the number of countable models

Morley’s theorem states that the number of non-isomorphic countable models of a
complete countable first-order theory in a countable language is ℵ_{0}or ℵ

_{1}or 2

^{ℵ0}. Vaught’s conjecture remains one of the most important open problems in Model Theory, asking whether ℵ

_{1}can be omitted in the conclusion of Morley’s theorem. Even though Vaught’s conjecture is trivially false in second-order logic, no result was known regarding Morley’s trichotomy for second-order logic. We shall show using forcing, large cardinals and descriptive set theory that the second-order version of Morley’s theorem is undecidable.

### 14:30

####
Andrew Marks and Spencer Unger^{﹡}:
Flows on the torus

In joint work with Andrew Marks, we gave a constructive solution to
Tarski's circle squaring problem. In particular, we showed that a
disk and a square with the same area are equidecomposible using
translations. One important innovation of the proof was to
construct a real valued flow from the disk to the square. The notion
of flow that we use comes from the study of networks and is related
to max flow-min cut. In this talk, I will sketch a simpler
construction of a real-valued flow from the disk to the square, which
is joint work with Andrew Marks. Using discrepancy estimates due to
Laczkovich, this argument works for sets whose boundary has small
upper Minkowski dimension. I will also mention ongoing work with
Anton Bernshteyn and Anush Tserunyan where we construct a large and
diverse collection of flows under the same assumptions.