# Sebastien Vasey: A proof of Shelah’s eventual categoricity conjecture in universal classes

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/

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Model Theory Seminar

Sebastien Vasey

CMU
Title: A proof of Shelahs 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 Morleys
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 Shelahs structure theory of univesal classes to remove the
amalgamation assumption.

Date: Monday, March 14, 2016
Time: 5:00 – 6:30 PM
Location: Wean 7201

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Model Theory Seminar

Sebastien Vasey

CMU
Title: A proof of Shelahs 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 Morleys
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 Shelahs structure theory of univesal classes to remove the
amalgamation assumption.

Date: Monday, March 21, 2016
Time: 5:00 – 6:30 PM
Location: Wean 7201
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Model Theory Seminar

Sebastien Vasey

CMU
Title: A proof of Shelahs 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 Morleys
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 Shelahs 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 Shelahs eventual categoricity conjecture in universal
classes, Part 4

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 Morleys
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 Shelahs structure theory of univesal classes to remove the
amalgamation assumption.

Date: Monday, April 4, 2016
Time: 5:00 – 6:30 PM
Location: Wean 7201