The seminar will continue to meet on Mondays in WeH 8201 at 5PM, the talks usually last 90 minutes.

Marcos Mazari Armida will give at least three talks, introducing Shelah’s good frames which the generalization to Abstract Elementary Classes of forking, he will focus on obtaining exists theorem of models when model theoretic assumptions will be replacing rather article non-ZFC axioms used by Shelah.

Rami Grossberg.

Marcos Mazari Armida

**Carnegie Melllon University**

**Title: **Introduction to good frames in Abstract Elementary Classes, Part 1

**Abstract: **

The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2

[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

**Date: **Monday , September 18, 2017.

**Time: **5:00 pm

**Location: **Wean Hall 8201

Marcos Mazari Armida

**Carnegie Melllon University**

**Title: **Introduction to good frames in Abstract Elementary Classes, Part 2

**Abstract: **

The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:

[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2

[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

**Date: **Monday , September 25, 2017.

**Time: **5:00 pm

**Location: **Wean Hall 8201

Marcos Mazari Armida

**Carnegie Melllon University**

**Title: **Introduction to good frames in Abstract Elementary Classes, Part 3

**Abstract: **

The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:

[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2

[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

**Date: **Monday , October 2, 2017.

**Time: **5:00 pm

**Location: **Wean Hall 8201