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## Upcoming talks and events:

# Why is Prikry forcing subcomplete?

Subcomplete forcing was introduced by Jensen as a class of forcings which do not add reals, but may change cofinalities to $\omega$, unlike proper forcing. In this talk I will show that Prikry forcing is subcomplete.

# Lattices of Elementary Substructures

The set of elementary substructures of a model of PA, under the inclusion relation, form a lattice. The lattice problem for models of PA asks which lattices can be represented as substructure lattices of some model of PA. This question dates back to Gaifman’s work on minimal types, which showed that the two element chain can be represented as a substructure lattice. Since then, there have been many important contributions to this problem, including by Paris, Wilkie, Mills, and Schmerl, though no complete picture has yet emerged. Studying this question involves knowledge of models of PA as well as some nontrivial lattice theory and combinatorics. In my talk, I will survey some of the major results and, if there’s time, give an idea of the techniques used to study this question.

# The definability of radicals in noncommutative rings

We will survey some results in the definability of radicals in rings, focusing on some recent results about noncommutative rings. In particular, we will show that there is no simple definition of the prime radical in the noncommutative setting, thus differentiating prime radicals in commutative/noncommutative rings.

# Title TBA

# Potential Cardinality for Countable First Order Theories

Give a theory $T$, understanding the countable model theory of $T$ has long been a topic of research. The number of countable models of $T$ is a classical but very coarse invariant, and this was refined significantly by Friedman and Stanley with the notion of Borel reductions.

Given theories $T_1$ and $T_2$, it is often straightforward to show that $T_1$ is Borel reducible to $T_2$. However, there are few tools to show that no such Borel reduction exists. Most of the existing tools only work when the isomorphism relation of one or both is particularly simple, or at least Borel.

We define the notion of “potential cardinality” of $T$, denoted $|T|$, as the number of formally consistent, possibly uncountable Scott sentences which imply $T$. It turns out that if $T_1$ Borel reduces to $T_2$, then $|T_1|$ is less than or equal to $|T_2|$. Additionally, it turns out that very frequently, $|T|$ can be computed and is not a proper class.

We use this idea to give a new class of examples of first-order theories whose isomorphism relations are neither Borel nor Borel complete. Along the way we answer an old question of Koerwien and new question of Laskowski and Shelah.

This is joint work with Douglas Ulrich and Chris Laskowski.

# Set Theory Day March 11, 2016 at GC

We announce a one-day conference, *Set Theory Day* on Friday, March 11, 2016, in celebration of the 50th birthday of Prof. Joel David Hamkins. The conference will take place at the CUNY Graduate Center and will last all day, with many of Prof. Hamkins’s former students giving talks on their work. All interested mathematicians are invited to attend. Further details will be posted here on nylogic.org as they become available.

# wPFA

We isolate a new forcing axiom, ${\rm wPFA}$, which is strictly between ${\rm BPFA}$ and ${\rm PFA}$. ${\rm wPFA}$ is equiconsistent with a remarkable cardinal, it implies the failure of $\square_{\omega_1}$, but it is compatible with $\square_\kappa$ for all $\kappa \geq \omega_2$. This is part of joint work with J. Bagaria and V. Gitman.

# No talks March 25

Friday, March 25 is a CUNY holiday, for Good Friday, and so there will be no talks that day.

# Title TBA

# NERDS on April 2, 2016

The Spring 2016 New England Recursion and Definability Seminar (“NERDS”) will take place on Saturday, April 2, 2016 at Springfield College, in Springfield, MA. Further details and abstracts of talks will be posted on nylogic.org as they become available.

# Title TBA

# Spring break April 22 & 29

CUNY’s spring break includes both Friday, April 22 and Friday, April 29, and so there will be no talks those days.