**CUNY Graduate Center**

**Room 4214-03**

**Wednesdays 6:30-7:30pm**

**May 24**

**Athar Abdul-Quader**, CUNY

**Defense of Dissertation: Interstructure Lattices and Types of Peano Arithmetic**

**3:30pm-5:30pm, Rm. 6300** (Note new time and room!)

**May 17**

**Somayeh Vojdani**, Cortica, New York

**Compact Groups in Non-Standard Models of Presburger Arithmetic**

We consider non-standard analogues of finite cyclic groups definable in Presburger arithmetic and give a description of their quotients by certain type-definable subgroups. These quotients are compact topological groups under the logic topology.

**May 10**

**Alf Dolich**, CUNY

**Recursive reducts of PA IV**

I will continue discussing work of Schmerl on generalizing Tennenbaum’s theorem. In particular I will show how to construct examples of theories that are (n+1)-rich and n-thin.

**May 3**

**Athar Abdul-Quader**, CUNY

**Minimal Elementary Extensions**

A recent paper by Schmerl characterizes those collections of sets which can be coded in a minimal elementary end extension. If M is a model of PA, X is such that there is some countably generated elementary end extension N with Cod(N / M) = X, and every set that is Pi^0_1 definable in (M, X) is a countable union of Sigma^0_1 definable sets, then M has a minimal elementary end extension N such that Cod(N / M) = X. I will go over the converse of this, as well as show an example of a model M and a collection X such that there is some N with Cod(N / M) = X but no minimal elementary end extension will code exactly those sets in X.

**April 26**

**Athar Abdul-Quader**, CUNY

**Interstructure lattices and Coded Sets III**

The lattice problem for models of PA has its origins in Gaifman’s work on minimal types. Gaifman showed that every model of PA has a conservative, minimal elementary end extension. This result showed a connection between interstructure lattices (in this case, the two element lattice), and the sets coded in an elementary end extension realizing that lattice (in this case, just the definable subsets). In subsequent years, work by Paris, Mills, and Schmerl studied this connection for other lattices. Most recently, Schmerl characterized the second order properties a collection of subsets of a model must have in order for there to be a minimal elementary end extension coding exactly those sets. In this series of talks, we will review the lattice problem for models of PA with the goal of studying when elementary end extensions realizing a particular interstructure lattice can code a prescribed collection of subsets of a model. Ultimately, we will show two main results: 1) for any countable model M, any finite distributive lattice D, and a collection X of subsets of M, M has an elementary end extension N such that Cod(N / M) = X if and only if M has an elementary end extension N such that the interstructure lattice Lt(N / M) = D and Cod(N / M) = X; and 2) for any model M, any finite Boolean algebra B, and a collection X of subsets of M, M has a minimal elementary end extension N such that Cod(N / M) = X if and only if M has an elementary end extension N such that Lt(N / M) = B and Cod(N / M) = X.

**April 19**

**Athar Abdul-Quader**, CUNY

**Interstructure lattices and Coded Sets II**

The lattice problem for models of PA has its origins in Gaifman’s work on minimal types. Gaifman showed that every model of PA has a conservative, minimal elementary end extension. This result showed a connection between interstructure lattices (in this case, the two element lattice), and the sets coded in an elementary end extension realizing that lattice (in this case, just the definable subsets). In subsequent years, work by Paris, Mills, and Schmerl studied this connection for other lattices. Most recently, Schmerl characterized the second order properties a collection of subsets of a model must have in order for there to be a minimal elementary end extension coding exactly those sets. In this series of talks, we will review the lattice problem for models of PA with the goal of studying when elementary end extensions realizing a particular interstructure lattice can code a prescribed collection of subsets of a model. Ultimately, we will show two main results: 1) for any countable model M, any finite distributive lattice D, and a collection X of subsets of M, M has an elementary end extension N such that Cod(N / M) = X if and only if M has an elementary end extension N such that the interstructure lattice Lt(N / M) = D and Cod(N / M) = X; and 2) for any model M, any finite Boolean algebra B, and a collection X of subsets of M, M has a minimal elementary end extension N such that Cod(N / M) = X if and only if M has an elementary end extension N such that Lt(N / M) = B and Cod(N / M) = X.

**April 5**

**Athar Abdul-Quader**, CUNY

**Interstructure lattices and Coded Sets I**

The lattice problem for models of PA has its origins in Gaifman’s work on minimal types. Gaifman showed that every model of PA has a conservative, minimal elementary end extension. This result showed a connection between interstructure lattices (in this case, the two element lattice), and the sets coded in an elementary end extension realizing that lattice (in this case, just the definable subsets). In subsequent years, work by Paris, Mills, and Schmerl studied this connection for other lattices. Most recently, Schmerl characterized the second order properties a collection of subsets of a model must have in order for there to be a minimal elementary end extension coding exactly those sets. In this series of talks, we will review the lattice problem for models of PA with the goal of studying when elementary end extensions realizing a particular interstructure lattice can code a prescribed collection of subsets of a model. Ultimately, we will show two main results: 1) for any countable model M, any finite distributive lattice D, and a collection X of subsets of M, M has an elementary end extension N such that Cod(N / M) = X if and only if M has an elementary end extension N such that the interstructure lattice Lt(N / M) = D and Cod(N / M) = X; and 2) for any model M, any finite Boolean algebra B, and a collection X of subsets of M, M has a minimal elementary end extension N such that Cod(N / M) = X if and only if M has an elementary end extension N such that Lt(N / M) = B and Cod(N / M) = X.

**March 29**

**Kameryn Williams**, CUNY

**Non-hyperfinite countable equivalence relations III**

I will present an argument of the Slaman–Steel theorem that the equivalence relation $E_\infty$ is not hyperfinite. Time permitting, I will explain a potential connection between this argument and the classification problem for models of arithmetic.

**March 22**

**Kameryn Williams**, CUNY

**Non-hyperfinite countable equivalence relations II**

I will present an argument of the Slaman–Steel theorem that the equivalence relation $E_\infty$ is not hyperfinite. Time permitting, I will explain a potential connection between this argument and the classification problem for models of arithmetic.

**March 15**

**Kameryn Williams**, CUNY

**Non-hyperfinite countable equivalence relations I**

I will present an argument of the Slaman–Steel theorem that the equivalence relation $E_\infty$ is not hyperfinite. Time permitting, I will explain a potential connection between this argument and the classification problem for models of arithmetic.

**March 8**

**Corey Switzer**, CUNY

**Recursively Saturated, Rather Classless Models of PA in the Constructible Universe**

In 1977 Kaufmann introduced the notion of rather classless models of PA and showed, under the assumption $\diamondsuit$, that every consistent extension of PA has an $\aleph_1$-like , recursively saturated, rather classless model. Shortly thereafter Shelah eliminated the assumption of $\diamondsuit$ by an absoluteness argument. In this talk I will present on work of Schmerl’s guided by the question of which cardinals can replace $\aleph_1$ in the Kaufmann-Shelah theorem. We show that assuming $V=L$ the question has a complete answer. The main result is the following: Assume $V=L$. Then, for any consistent extension T of PA and and uncountable cardinal $\kappa$, there is a $\kappa$-like, recursively saturated, rather classless model of T if and only if $cf(\kappa) > \aleph_0$ and $\kappa$ is not weakly compact. Central to these arguments are the use of satisfaction classes and they will be discussed as well.

**March 1**

**Corey Switzer**, CUNY

**Recursively Saturated, Rather Classless Models of PA in the Constructible Universe**

In 1977 Kaufmann introduced the notion of rather classless models of PA and showed, under the assumption $\diamondsuit$, that every consistent extension of PA has an $\aleph_1$-like , recursively saturated, rather classless model. Shortly thereafter Shelah eliminated the assumption of $\diamondsuit$ by an absoluteness argument. In this talk I will present on work of Schmerl’s guided by the question of which cardinals can replace $\aleph_1$ in the Kaufmann-Shelah theorem. We show that assuming $V=L$ the question has a complete answer. The main result is the following: Assume $V=L$. Then, for any consistent extension T of PA and and uncountable cardinal $\kappa$, there is a $\kappa$-like, recursively saturated, rather classless model of T if and only if $cf(\kappa) > \aleph_0$ and $\kappa$ is not weakly compact. Central to these arguments are the use of satisfaction classes and they will be discussed as well.

**February 22**

**Michał Tomasz Godziszewski**, University of Warsaw

**Standard Models of Arithmetic**

The purpose of the talk is to give the proof of the following characterization of ${\rm ZF}$-standard models of ${\rm PA}$ (i.e. models being the standard model of arithmetic in the sense of some model of set theory) given by Ali Enayat in his paper on standard models of arithmetic:

let $T$ be a recursively axiomatizable extension of ${\rm ZF}$. Then for any nonstandard model $M\models {\rm PA}$ with a countable cofinality the following are equivalent:

(i) $M$ is a $T$-standard model of ${\rm PA}$

(ii) $M$ is recursively saturated model of ${\rm PA} + A_T$, where $A_T$ is a recursive axiomatization of the arithmetical consequences of $T$.

**February 15**

**Alf Dolich**, CUNY

**Recursive Reducts of PA III**

I will continue my overview of Schmerl’s work on generalizations of Tennenbaum’s theorm to general reducts of Peano Arithmetic.