**CUNY Graduate Center**

**Room 4214.03**

**Tuesdays 6:30pm-8pm**

**Organized by Roman Kossak**

**Calendar**

**October 23**

**Roman Kossak**,
CUNY

**Nonstandard definability**

**Abstract**

In 1989, Stuart Smith proved that for every full truth predicate T on a nonstandard model M of PA, there is a nonstandard formula that, according to T, defines an undefinable class of M. Recently, Bartosz Wcisło improved this by showing that there is also a nonstandard formula that defines an inductive partial truth predicate. The proof uses the technique of disjunctions with stopping conditions. I will discuss the proof, and, if time permits, I will also talk about the end extension problem for full truth predicates.

**October 16**

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

**Kernels of digraphs and truth classes**

**Abstract**

I will talk about a recent paper by Jim Schmerl which provides an alternative proof that every resplendent model of PA has a full truth class. The proof, surprisingly, boils down to studying kernels of digraphs. A kernel of a digraph is a set K such that for any a, b in K, (a, b) is not an edge, and for any a not in K, there is some b in K such that (a, b) is an edge. Schmerl shows that one can view the result that resplendent models have full truth classes as a particular instance of the result that resplendent digraphs which have local finite height have kernels.

**October 9**

**Corey Switzer**,
CUNY

**Applications of $(\mathcal L, n)$ Models, Part II**

**Abstract**

We will continue discussing Shelah's technique of $(\mathcal L, n)$ models and their applications to independence results in PA. This includes a new proof of Paris-Harrington and an example of a concrete true but unprovable $\Pi^0_1$ sentence. Time permitting we'll also discuss the connection of these ideas to Wilkie's theorem on Why PA? and Kripke's notion of fulfillment.

**September 25**

**Corey Switzer**,
CUNY

**Applications of $(\mathcal L, n)$ Models**

**Abstract**

We'll discuss Shelah's technology of $(\mathcal L, N)$ models and its applications to independence results in PA. This includes an alternative proof of the Paris-Harrington theorem and a sharpening to a $\Pi^0_1$ true but unprovable statement of some mathematical interest. Time permitting, we'll also connect these ideas to Kripke's notion of Fulfillment and Wilkie's theorem on Why PA.

**Previous Semesters**