Spring 2018

**CUNY Graduate Center**

**Room 4213.03 (Math Thesis Room)**

**Wednesdays 6:30-8:00pm**

**Organized by Alf Dolich and and Kameryn Williams**

**February 7**

** Kameryn Williams**, CUNY

**Forcing over arithmetic: a second-order approach**

Continuing a theme of previous talks in this seminar, I will talk about forcing over models of arithmetic. I will present a framework for forcing over models of $\mathsf{ACA}_0$, generalizing the approach (as seen in e.g. Kossak and Schmerl’s book) of looking at definable sets over a model of PA. This is analogous to the approach within set theory of developing class forcing in GBC, rather than using definable classes over ZFC. The main result I will present is that forcing preserves the axioms of $\mathsf{ACA}_0$ and, indeed, $\Pi^1_k\text{-}\mathsf{CA}$.

This is part of joint work with Corey Switzer about generalizing Kossak and Schmerl’s results about perfect generics.

**February 14**

** Kameryn Williams**, CUNY

**Forcing over arithmetic: a second-order approach**

This is a continuation of the previous week’s talk.

Continuing a theme of previous talks in this seminar, I will talk about forcing over models of arithmetic. I will present a framework for forcing over models of $\mathsf{ACA}_0$, generalizing the approach (as seen in e.g. Kossak and Schmerl’s book) of looking at definable sets over a model of PA. This is analogous to the approach within set theory of developing class forcing in GBC, rather than using definable classes over ZFC. The main result I will present is that forcing preserves the axioms of $\mathsf{ACA}_0$ and, indeed, $\Pi^1_k\text{-}\mathsf{CA}$.

This is part of joint work with Corey Switzer about generalizing Kossak and Schmerl’s results about perfect generics.

**February 21**

**Corey Switzer**, CUNY

**Forcing Over Arithmetic: Arboreal Forcings**

Building off the ideas set up in the previous two talks on forcing over models of ACA_0 we develop a general framework for arboreal forcings over models of arithmetic: that is forcing with trees. Previously Kossak and Schmerl have developed this in the case of Sacks’ forcing though we show that similar ideas work for a much wider range of families of trees. In particular we will consider Hechler forcing and Laver forcing in this framework. On the one hand these forcings are similar to hyperclass forcing in set theory as conditions are classes, on the other hand there is a strong connection with the forcings to add reals that our arboreal forcings come from. Similar to the case in set theory, we will show that Hechler forcing adds a Cohen generic but Laver forcing does not. However, Laver forcing twice does.

This is part of joint work with Kameryn Williams generalizing Kossak and Schmerl’s results about perfect generics.

**February 28**

**Corey Switzer**, CUNY

**Forcing Over Arithmetic: Arboreal Forcings**

This is a continuation of the previous week’s talk.

Building off the ideas set up in the previous two talks on forcing over models of ACA_0 we develop a general framework for arboreal forcings over models of arithmetic: that is forcing with trees. Previously Kossak and Schmerl have developed this in the case of Sacks’ forcing though we show that similar ideas work for a much wider range of families of trees. In particular we will consider Hechler forcing and Laver forcing in this framework. On the one hand these forcings are similar to hyperclass forcing in set theory as conditions are classes, on the other hand there is a strong connection with the forcings to add reals that our arboreal forcings come from. Similar to the case in set theory, we will show that Hechler forcing adds a Cohen generic but Laver forcing does not. However, Laver forcing twice does.

This is part of joint work with Kameryn Williams generalizing Kossak and Schmerl’s results about perfect generics.

**March 7**

**Seminar cancelled**

**March 14**

**Seminar cancelled**

**March 21**

** Seminar cancelled due to inclement weather!**

**Alf Dolich**, CUNY

**Recursive Functions and Existentially Closed Structures**

I will begin to discuss a recent paper of Jerabek which investigates the relationships between various conditions implying essential undecidability. This investigation uses a host of interesting tools from contemporary model theory.