Degrees of members of Π classes Ahmet C¸ evik

116 Pages·2014·0.64 MB·English

Degrees of members of Π0 classes
1
Ahmet C¸evik
Submitted in accordance with the requirements
for the degree of Doctor of Philosophy
The University of Leeds
School of Mathematics
August 2014
The candidate confirms that the work submitted is his own and that appropri-
atecredithasbeengivenwherereferencehasbeenmadetotheworkofothers.
This copy has been supplied on the understanding that it is copyright mate-
rial and that no quotation from the thesis may be published without proper
acknowledgement.
(cid:13)c2014 The University of Leeds and Ahmet C¸evik

Acknowledgments
I would like to express my deepest gratitude to Andy Lewis for his great effort
in supervising this thesis and for all his support, for having had patience with
me and for his generosity through many years. I truly feel lucky to have a
mentor like him as it has been a great experience to have been able to discuss
mathematicalissueswithhimandtolearnsomanythingsfromhissupervision.
I hope we have a chance to continue our discussions in the future as well. I
would like to thank Barry Cooper a lot for always giving me useful advice
regarding academic issues whenever I needed help from a senior professional
and for encouraging me to participate in conferences, workshops and so forth.
I hope we stay in contact.
I would like to give a very special thank you to my external examiner
Anton´ın Kuˇcera for his detailed comments and carefully pointing out his sug-
gestions which have been very useful for improving the quality of this thesis.
I would also like to thank my internal examiner Peter Schuster for reading
my correction report, for scheduling my viva and making me feel comfortable
during that time. I should thank our postgraduate research secretary Jeanne
Shuttleworthforhelpinguswithourproblemsregardingformalproceduresand
for her patience.
IwouldliketothankmyfriendsfromLeedsincludingAndr´esArandaL´opez,
JoelRonnieNagloo, MichaelToppel,PedroFranciscoValenciaVizca´ıno, Rizos
Sklinos, SerkanAydın and many others formaking Leedsabetter place tolive
and for their company.
Lastbutnotleast,Iwouldtothankmyfamilyfortheircontinuoussupport,
maybe not regarding mathematics but anything else.
iii
iv Acknowledgments
This thesis is dedicated to my parents and my sister.
Abstract
In this thesis we study Turing degrees of members of Π0 classes. We give two
1
introductorychaptersandthenthreemainchapterswhichincludenewresults.
Inthefirstchapterwegivesomestandardbackgroundforrecursiontheory,
and we give an introduction to Π0 classes in the second chapter.
1
The third chapter will be on the published work [1]. We show that for any
degree a≥0′, if a Π0 class P contains members of every degree b such that
1
b′ =a, thenP containsmembersofeverydegree. Alocalversionofthisresult
is also given. That is, when a is also Σ0, it suffices in the hypothesis to have a
2
member of every ∆0 degree b such that b′ =a. This result extends the Low
2
Antibasis Theorem given in Kent and Lewis [2].
The fourth chapter has three subsections. The first subsection concerns an
observation, which may be seen as a cupping non-basis analogue of Jockusch
and Soare’s capping basis theorem: We show that there exists a non-empty
Π0 class with no recursive member, such that no join of two sets in the class
1
computes ∅′. The second one contains the principal result of the chapter,
which concerns the relationship between the join property and the members of
Π0 classes. We show that there exists a non-empty Π0 class with no recursive
1 1
member, for which it also holds that no member satisfies the join property.
Thirdsubsectioncontainssomefutureworkwherewegivesomeopenquestions
about the relation between minimal covers and Π0 classes, and also about the
1
relation between minimal covers and PA degrees.
In the fifth chapter we study the degree spectrum properties of a special
kind of Π0 classes that we introduce, so called Π0 choice classes. A Π0 choice
1 1 1
class is a Π0 class such that no two members have the same Turing degree.
1
v
vi Abstract
Consideringthisrestrictedclassleadsustosomeinterestingantibasistheorems
and technically innovative constructions.
Contents
Acknowledgments iii
Abstract v
Contents vii
Abbreviations ix
List of Figures xi
1 Background on Recursion Theory 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Turing degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Construction methods . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Jump classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Degrees of Peano Arithmetic and Π0 Classes 29
1
2.1 Cantor Space and Topology . . . . . . . . . . . . . . . . . . . . 30
2.2 Axiomatizable theories . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Basis theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 PA Degrees and Π0 Classes . . . . . . . . . . . . . . . . . . . . 40
1
2.5 Variants of Π0 classes . . . . . . . . . . . . . . . . . . . . . . . 43
1
vii
viii Contents
3 Antibasis theorems and jump inversion 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Modifying σ(i,j,τ) . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 First theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Second theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Join Property and effectively closed sets 57
4.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Cupping Non-basis theorem . . . . . . . . . . . . . . . . . . . . 61
4.2.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Join property and Π0 classes . . . . . . . . . . . . . . . . . . . 63
1
4.3.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Choice Classes 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Properties of (P ,<) . . . . . . . . . . . . . . . . . . . . . . . . 76
c
5.3 Decidability of the ∃-theory of (P ,<) . . . . . . . . . . . . . . 91
c
5.4 Choice invisible degrees . . . . . . . . . . . . . . . . . . . . . . 93
5.4.1 Random sets and Π0 choice classes . . . . . . . . . . . . 96
1
Bibliography 99
Abbreviations
ANR Array non-recursive
DNR Diagonally non-recursive
domf Domain of the function f
FPF Fixed point free
g.l.b. Greatest lower bound
l.u.b. Least upper bound
PA Peano arithmetic
poset Partially ordered set
r.e. Recursively enumerable
ZFC Zermelo-Fraenkel set theory with the Axiom of Choice
ix
x Abbreviations