Interstructure Lattices and Types of Peano Arithmetic

78 Pages·2017·0.52 MB·English

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Dissertations, Theses, and Capstone Projects CUNY Graduate Center
9-2017
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Athar Abdul-Quader
The Graduate Center, City University of New York
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Interstructure Lattices and Types of Peano Arithmetic
by
Athar Abdul-Quader
A dissertation submitted to the Graduate Faculty in Mathematics in partial
fulfillment of the requirements for the degree of Doctor of Philosophy, The
City University of New York
2017
ii
(cid:13)c 2017
Athar Abdul-Quader
All Rights Reserved
iii
This manuscript has been read and accepted by the Graduate Faculty in
Mathematics in satisfaction of the dissertation requirement for the degree of
Doctor of Philosophy.
Roman Kossak
Date Chair of Examining Committee
Ara Basmajian
Date Executive Officer
Roman Kossak
Alfred Dolich
Russell Miller
Philipp Rothmaler
Supervisory Committee
The City University of New York
iv
Abstract
Interstructure Lattices and Types of Peano Arithmetic
by
Athar Abdul-Quader
Advisor: Professor Roman Kossak
ThecollectionofelementarysubstructuresofamodelofPAformsalattice,and
is referred to as the substructure lattice of the model. In this thesis, we study
substructure and interstructure lattices of models of PA. We apply techniques
used in studying these lattices to other problems in the model theory of PA.
In Chapter 2, we study a problem that had its origin in Simpson ([Sim74]),
who used arithmetic forcing to show that every countable model of PA has an
expansion to PA∗ that is pointwise definable. Enayat ([Ena88]) later showed
that there are 2ℵ0 models with the property that every expansion to PA∗ is
pointwise definable. In this Chapter, we use techniques involved in represen-
tations of lattices to show that there is a model of PA with this property which
contains an infinite descending chain of elementary cuts.
In Chapter 3, we study the question of when subsets can be coded in ele-
mentary end extensions with prescribed interstructure lattices. This problem
originated in Gaifman [Gai76], who showed that every model of PA has a con-
servative, minimal elementary end extension. That is, every model of PA has
a minimal elementary end extension which codes only definable sets. Kossak
v
and Paris [KP92] showed that if a model is countable and a subset X can be
coded in any elementary end extension, then it can be coded in a minimal
extension. Schmerl ([Sch14] and [Sch15]) extended this work by considering
which collections of sets can be the sets coded in a minimal elementary end
extension. In this Chapter, we extend this work to other lattices. We study
two questions: given a countable model M, which sets can be coded in an el-
ementary end extension such that the interstructure lattice is some prescribed
finite distributive lattice; and, given an arbitrary model M, which sets can be
coded in an elementary end extension whose interstructure lattice is a finite
Boolean algebra?
Acknowledgments
First I owe my sincerest gratitude to my advisor, Roman Kossak, for his pa-
tience, kindness, and consistent words of encouragement. Roman was an ex-
cellent teacher and I always feel that I come away learning something new
whenever I talk to him.
It is hard to put into words the mathematical debt I owe Jim Schmerl.
Every result in this thesis originated by reading his papers and some came
directly from talking or emailing with him.
There were many times in the “Logic Office” that Alf Dolich answered
questions from Roman and me about model theory and inspired some idea
or other that ended up going into the thesis. He of course is the inspiration
behind the question about “Dolich sets” in Chapter 2. It has been extremely
helpful to have Alf’s perspective on problems in PA.
I am deeply indebted to the logic faculty at CUNY. Between classes and
seminars, it was an honor and a pleasure to learn from Russell Miller, Gunter
Fuchs, Philipp Rothmaler and Vika Gitman.
TomyfellowMOPA-ians: KamerynWilliams,SimonHeller,KerryOjakian,
vi
vii
Erez Shochat, Whanki Lee, and Corey Switzer. Thank you for listening to me
ramble on about lattices for years by now. I learned a lot from you all and
I could consistently count on your questions whenever there was an idea I
needed to further develop or a different perspective on a problem I needed to
see.
I am incredibly grateful to my family for all the love and support they
have given me over the years. I am so lucky to be married to my best friend,
Sharmin Ahmed. My entire life I had to look up to my siblings, Azhar and
Ayesha. Thank you both for not getting PhDs so I can finally outshine you
two. (More seriously: your lives and successes have always been an inspiration
for me.) My mother, Fauzia, is the kindest person I know. My love of math
started when I was young because of her. She has always believed in me more
than I ever believed in myself.
In1972Dr. MohammedAbdul-QuaderleftIndiaforanewlifewitharound
$8 in his pocket. He worked harder than anyone I ever knew, raising his entire
family of 9 sisters and 2 brothers after his father met his demise. When I
decided to leave my job in order to attend graduate school, he took me back
into his home with no questions asked. On March 19, 2017, he returned to his
Lord. Anything I could say about him in this space would be inadequate, so
let me close with this: Dad, I miss you, I love you, and I dedicate this work
to your memory. May your soul be at peace with the Creator of Peace.
Contents
Contents viii
1 Introduction 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Enayat Models 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Enayat Models . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Enayat Models with Infinite Descending Sequences of Elemen-
tary Submodels . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Dolich Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Diversity in Elementary Substructures . . . . . . . . . . . . . 45
2.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Coded Sets and Interstructure Lattices 50
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Coding Sets in Elementary Extensions . . . . . . . . . . . . . 51
viii
CONTENTS ix
3.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 62