Sebastien Vasey will discuss his solution to Shelah’s categoricity

conjecture for universal AECs. Unlike previous approximations, this is

without additional model theoretic or set-theoretic assumptions, the proof

is entirely in ZFC. The relevant papers are #8, #11 and #16 from

http://math.cmu.edu/~svasey/

Model Theory Seminar

Sebastien Vasey

CMU

Title: A proof of Shelahs eventual categoricity conjecture in universal

classes, Part 1

Abstract: Abstract elementary classes (AECs) are an axiomatic framework

encompassing classes of models of an $\mathbb{L}_{\infty, \omega}$ theory,

as well as numerous algebraic examples. They were introduced by Saharon

Shelah forty years ago. Shelah focused on generalizations of Morleys

categoricity theorem and conjectured the following eventual version: An

AEC categorical in a high-enough cardinal is categorical on a tail of

cardinals. I will present my proof of the conjecture for universal

classes. They are a special case of AECs (studied by Shelah in a milestone

1987 paper) corresponding to classes of models of a universal

$\mathbb{L}_{\infty, \omega}$ theory.

I will initially discuss the proof with the additional assumption that the

universal class satisfies the amalgamation property. In this case, the

argument generalizes to AECs which have amalgamation, are tame (a locality

property for orbital types isolated by Grossberg and VanDieren), and have

primes over sets of the form $Ma$. Time permitting, I will discuss how to

use Shelahs structure theory of univesal classes to remove the

amalgamation assumption.

Date: Monday, March 14, 2016

Time: 5:00 – 6:30 PM

Location: Wean 7201

Model Theory Seminar

Sebastien Vasey

CMU

Title: A proof of Shelahs eventual categoricity conjecture in universal

classes, Part 2

Abstract: Abstract elementary classes (AECs) are an axiomatic framework

encompassing classes of models of an $\mathbb{L}_{\infty, \omega}$ theory,

as well as numerous algebraic examples. They were introduced by Saharon

Shelah forty years ago. Shelah focused on generalizations of Morleys

categoricity theorem and conjectured the following eventual version: An

AEC categorical in a high-enough cardinal is categorical on a tail of

cardinals. I will present my proof of the conjecture for universal

classes. They are a special case of AECs (studied by Shelah in a milestone

1987 paper) corresponding to classes of models of a universal

$\mathbb{L}_{\infty, \omega}$ theory.

I will initially discuss the proof with the additional assumption that the

universal class satisfies the amalgamation property. In this case, the

argument generalizes to AECs which have amalgamation, are tame (a locality

property for orbital types isolated by Grossberg and VanDieren), and have

primes over sets of the form $Ma$. Time permitting, I will discuss how to

use Shelahs structure theory of univesal classes to remove the

amalgamation assumption.

Date: Monday, March 21, 2016

Time: 5:00 – 6:30 PM

Location: Wean 7201

Model Theory Seminar

Sebastien Vasey

CMU

Title: A proof of Shelahs eventual categoricity conjecture in universal

classes, Part 3

Abstract: Abstract elementary classes (AECs) are an axiomatic framework

encompassing classes of models of an $\mathbb{L}_{\infty, \omega}$ theory,

as well as numerous algebraic examples. They were introduced by Saharon

Shelah forty years ago. Shelah focused on generalizations of Morleys

categoricity theorem and conjectured the following eventual version: An

AEC categorical in a high-enough cardinal is categorical on a tail of

cardinals. I will present my proof of the conjecture for universal

classes. They are a special case of AECs (studied by Shelah in a milestone

1987 paper) corresponding to classes of models of a universal

$\mathbb{L}_{\infty, \omega}$ theory.

I will initially discuss the proof with the additional assumption that the

universal class satisfies the amalgamation property. In this case, the

argument generalizes to AECs which have amalgamation, are tame (a locality

property for orbital types isolated by Grossberg and VanDieren), and have

primes over sets of the form $Ma$. Time permitting, I will discuss how to

use Shelahs structure theory of univesal classes to remove the

amalgamation assumption.

Date: Monday, March 28, 2016

Time: 5:00 – 6:30 PM

Location: Wean 7201

Model Theory Seminar

Sebastien Vasey

CMU

Title: A proof of Shelahs eventual categoricity conjecture in universal

classes, Part 4

encompassing classes of models of an $\mathbb{L}_{\infty, \omega}$ theory,

as well as numerous algebraic examples. They were introduced by Saharon

Shelah forty years ago. Shelah focused on generalizations of Morleys

categoricity theorem and conjectured the following eventual version: An

AEC categorical in a high-enough cardinal is categorical on a tail of

cardinals. I will present my proof of the conjecture for universal

classes. They are a special case of AECs (studied by Shelah in a milestone

1987 paper) corresponding to classes of models of a universal

$\mathbb{L}_{\infty, \omega}$ theory.

universal class satisfies the amalgamation property. In this case, the

argument generalizes to AECs which have amalgamation, are tame (a locality

property for orbital types isolated by Grossberg and VanDieren), and have

primes over sets of the form $Ma$. Time permitting, I will discuss how to

use Shelahs structure theory of univesal classes to remove the

amalgamation assumption.

Date: Monday, April 4, 2016

Time: 5:00 – 6:30 PM

Location: Wean 7201