**CUNY Graduate Center**

**Room 4214.03**

**Wednesdays 6:30pm-8pm**

**Organized by Roman Kossak**

**Calendar**

**April 10**

**Erez Shochat**,
St. Francis College

**Introduction to Loeb Measure**

**Abstract**

In this talk we will outline results and facts from nonstandard analysis and introduce the concept of Loeb Measure.

**April 3**

**Michał Tomasz Godziszewski**,
University of Warsaw

**$\Pi^0_1$-computable quotient presentations of nonstandard models of arithmetic**

**Abstract**

A computable quotient presentation of a mathematical structure $\mathcal A$ consists of a computable structure on the natural numbers $\langle \mathbb{N},\star,\ast,\dots \rangle$, meaning that the operations and relations of the structure are computable, and an equivalence relation $E$ on $\mathbb{N}$, not necessarily computable but which is a congruence with respect to this structure, such that the quotient $\langle \mathbb{N},\star,\ast,\dots \rangle$ is isomorphic to the given structure $\mathcal{A}$. Thus, one may consider computable quotient presentations of graphs, groups, orders, rings and so on.

A natural question asked by B. Khoussainov in 2016, is if the Tennenbaum Thoerem extends to the context of computable presentations of nonstandard models of arithmetic. In a joint work with J.D. Hamkins we have proved that no nonstandard model of arithmetic admits a computable quotient presentation by a computably enumerable equivalence relation on the natural numbers.

However, as it happens, there exists a nonstandard model of arithmetic admitting a computable quotient presentation by a co-c.e. equivalence relation. Actually, there are infinitely many of those. The idea of the proof consists in simulating the Henkin construction via finite injury priority argument. What is quite surprising, the construction works (i.e. injury lemma holds) by Hilbert's Basis Theorem. During the talk I'll present ideas of the proof of the latter result, which is joint work with T. Slaman and L. Harington.

**March 27**

**Simon Heller**,
CUNY

**Definable sets in Presburger arithmetic**

**Abstract**

I will give a proof of the Ginsburg-Spanier theorem, as well as the nice corollary that multiplication is not definable in Presburger.

**March 20**

**Roman Kossak**,
CUNY

**Complexity of the isomorphism relation for finitely generated models of PA**

**Abstract**

I present a proof, from a joint paper with Sam Coskey, that the isomorphism relation for finitely generated models of PA is not smooth.

**March 13**

**Athar Abdul-Quader**,
SUNY Purchase College

**Mathias genericity in second order models of arithmetic**

**Abstract**

I will continue speaking about some results in the reverse mathematics of combinatorial principles which utilize Mathias forcing.

**March 6**

**Athar Abdul-Quader**,
SUNY Purchase College

**Mathias genericity in second order models of arithmetic**

**Abstract**

I will review some results about Mathias forcing over models of RCA0. This method has recently been used by Slaman and Yokoyama to prove conservativity results for Ramsey's Theorem for pairs. I will mostly discuss just the method, rather than this new result in particular.

**February 27**

**Corey Switzer**,
CUNY

**Sets at arm's length**

**Abstract**

We will present the following striking theorem, due to Towsner: If $(M, \mathcal X) \models \mathsf{RCA}_0 + I\Sigma_n$ is countable, then for *any* (!!!) $W \subseteq M$ there is a countable expansion with the same first order part, $(M, \mathcal Y)$, where $\mathcal X \subseteq \mathcal Y$ so that $(M, \mathcal Y) \models \mathsf{RCA}_0 + I\Sigma_n$ and the set $W$ is $\Delta_{n+1}$ definable. Time permitting, we will then sketch some nice applications of this theorem to the study of conservative extensions of fragments of arithmetic.

**February 6**

**Whan Ki Lee**,
CUNY

**$\kappa$-like models**

**Abstract**

This is a continuation of a talk from last semester.

A model $(M, < ,\ldots)$ is said to be $\kappa$-like if $|M| = \kappa$ but for all $a \in M$, $|\{x \in M \mid x < a\}| < \kappa$. Based on the paper, the theory of $\kappa$-like models of arithmetic by R. Kaye, we will identify some axiom schemes true in such models of $I\Delta_0$ and investigate their interesting properties.

**Previous Semesters**