BEYOND THE ARITHMETIC - Cornell University

322 Pages·2005·2.14 MB·English

BEYOND THE ARITHMETIC
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Antonio Montalb´an
August 2005
This document is in the public domain.
BEYOND THE ARITHMETIC
Antonio Montalb´an, Ph.D.
Cornell University 2005
Various results in different areas of Computability Theory are proved.
FirstweworkwiththeTuringdegreestructure, provingsomeembeddablityand
decidability results. To cite a few: we show that every countable upper semilattice
containing a jump operation can be embedded into the Turing degrees, of course,
preserving jump and join; we show that every finite partial ordering labeled with
the classes in the generalized high/low hierarchy can be embedded into the Turing
degrees; we show that every generalized high degree has the complementation
property; and we show that if a Turing degree a is either 1-generic and ∆0, 2-
1
generic and arithmetic, n-REA, or arithmetically generic, then the theory of the
partial ordering of the Turing degrees below a is recursively isomorphic to true
first order arithmetic.
Second, we work with equimorphism types of linear orderings from the view-
points of Computable Mathematics and Reverse Mathematics. (Two linear or-
derings are equimorphic if they can be embedded in each other.) Spector proved
in 1955 that every hyperarithmetic ordinal is isomorphic to a recursive one. We
extend his result and prove that every hyperarithmetic linear ordering is equimor-
phic to a recursive one. From the viewpoint of Reverse Mathematics, we look at
the strength of Fra¨ıss´e’s conjecture. From our results, we deduce that Fra¨ıss´e’s
conjecture is sufficient and necessary to develop a reasonable theory of equimor-
phism types of linear orderings. Other topics we include in this thesis are the
following: we look at structures for which Arithmetic Transfinite Recursion is the
natural system to study them; we study theories of hyperarithmetic analysis and
present a new natural example; we look at the complexity of the elementary equiv-
alence problem for Boolean algebras; and we prove that there is a minimal pair of
Kolmogorov-degrees.
BIOGRAPHICAL SKETCH
I was born and raised in Montevideo, Uruguay. I started to take mathematics
seriously when I join the math Olympiads team of Uruguay, in 1994. I spent three
years with the math Olympiads solving fun and elementary math problems. With
them I got to travel to competitions and meet many people from all around Latin
America. In 1997, I entered the undergraduate program in Mathematics at the
Universidad de la Repu´blica in Montevideo, Uruguay. There I had Paula Severi
and Walter Ferrer as my advisers. They are the ones who introduced me to the
subject of logic. In 2000, I got my degree of Licenciado en Matem´aticas and came
to Cornell.
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ACKNOWLEDGEMENTS
I am very grateful to my thesis adviser Richard A. Shore for all his help during
my time here at Cornell. I could not have had a better adviser. He gave me many
interesting problems for me to look at, and he would listen to all my ideas and
read all my drafts carefully and comment on them. He advised me not only about
mathematics, but also about any other aspect of the academic life.
I also want to thank my co-authors Noam Greenberg, Barbara F. Csima and
Bjørn Kjos-Hanssen for letting me include the work we did together in this thesis.
It has been fun working with them.
Iwillnotbeabletolistallthepeoplethatsupportedmeinsomewayoranother
during the last five years. I will just list a few of them. On the one hand are the
people that I met here in Ithaca, without whom I would not have enjoyed my time
here as much. Among these people are all friends, my closer ones being Alison,
Cristina, Everilis, Fabio, Fernando, Gabor, Johnny, Jorge, Mercedes, Pedro, Rafael
and Yannet. I also want to mention Andr´es, Carmen, Marta, Marcelo and Ricardo,
whohelpedmeoutinmyfirstfewdaysinIthaca. Ontheotherhandarethepeople
athomethat, eventhoughwheremanythousandsofkilometersaway, wherealways
with me. These include my friends Alejandro and Aldo, and of course my family,
my mother, my father and my sister that were always there to support me.
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TABLE OF CONTENTS
1 General Introduction 1
1.1 Turing Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Reverse Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Computable Mathematics . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Effective Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Scattered linear orderings . . . . . . . . . . . . . . . . . . . . . . . 10
I Turing Degree structure 11
2 Embedding Jump upper semilattices into the Turing Degrees. 12
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The Main construction . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 The forcing notion. . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Preservation of nonorder. . . . . . . . . . . . . . . . . . . . . 18
2.2.3 R preserves the jump. . . . . . . . . . . . . . . . . . . . . . 20
G
2.3 Decidability results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Jump upper semilattices which support Jump Hierarchies. . . . . . 23
2.4.1 Pseudo-well orderings with Jump Hierarchies. . . . . . . . . 23
2.4.2 Partial upper semilattices with level function. . . . . . . . . 26
2.4.3 Well quasiorderings. . . . . . . . . . . . . . . . . . . . . . . 28
2.4.4 The decomposition of J. . . . . . . . . . . . . . . . . . . . . 30
2.4.5 Putting the pieces together. . . . . . . . . . . . . . . . . . . 31
2.5 Adding 0 to the Language . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.1 A negative answer. . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.2 A positive answer. . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 Uncountable jump upper semilattices . . . . . . . . . . . . . . . . . 37
2.6.1 A negative answer. . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.2 A positive answer . . . . . . . . . . . . . . . . . . . . . . . . 39
3 There is no order in the Generalized High/Low Hierarchy. 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 GH-posets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 The main Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 True Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Tree Constructions . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.3 Construction of B . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.4 Construction of A . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.5 Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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4 Generalized High degrees have the complementation property
(with Noam Greenberg and Richard A. Shore). 60
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Case One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Case Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Construction of the Tree . . . . . . . . . . . . . . . . . . . . 69
4.3.2 The Construction . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.4 Coding D into B . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Case Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.1 A minimal degree below D . . . . . . . . . . . . . . . . . . . 75
4.4.2 Construction I . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.3 Coding D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4.4 Construction II . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4.5 Verifications II . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Embeddingandcodingbelowa1-genericdegree(with Noam Green-
berg). 84
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2 1-Genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Slaman and Woodin Coding . . . . . . . . . . . . . . . . . . . . . . 88
5.2.1 Coding Countable Sets . . . . . . . . . . . . . . . . . . . . . 92
5.3 Interpreting Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 Lattice Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
II Reverse Mathematics 101
6 Equivalence between Frass’s conjecture and Jullien’s theorem. 102
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Signed trees and h-indecomposable linear orderings . . . . . . . . . 110
6.2.1 Signed trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2 H-indecomposable linear orderings . . . . . . . . . . . . . . . 112
6.2.3 WQO(ST) implies ATR . . . . . . . . . . . . . . . . . . . . 115
0
6.3 Finite decomposability . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.1 FINDEC and WQO(ST) . . . . . . . . . . . . . . . . . . . . . 119
6.3.2 Minimal decomposition . . . . . . . . . . . . . . . . . . . . . 122
6.4 Fra¨ıss´e’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.5 Jullien’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.1 Proof of the easy direction . . . . . . . . . . . . . . . . . . . 125
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6.5.2 Implications of JUL . . . . . . . . . . . . . . . . . . . . . . . 128
6.5.3 Minimal decomposition and the proof of Jullien’s theorem . 130
6.6 Extendibility of h-indecomposable linear orderings . . . . . . . . . . 134
6.6.1 Extendibility of ω∗ and (ω2)∗ . . . . . . . . . . . . . . . . . 134
6.6.2 Extendibility of 1+L+1 . . . . . . . . . . . . . . . . . . . 136
P
6.6.3 Extendibility of L . . . . . . . . . . . . . . . . . . . 137
m∈ω∗ m
6.6.4 One step iteration . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6.5 The complement of a linear ordering . . . . . . . . . . . . . 139
6.6.6 The linearization . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6.7 Extendibility of η . . . . . . . . . . . . . . . . . . . . . . . . 144
6.7 Another equivalent statement . . . . . . . . . . . . . . . . . . . . . 145
7 Indecomposable linear orderings and Theories of Hyperarithmetic
Analysis. 147
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.1.1 Indecomposability Statement . . . . . . . . . . . . . . . . . 149
7.1.2 Game statements . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1.3 The Jump Iteration statement . . . . . . . . . . . . . . . . . 153
7.1.4 Summary or results . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.5 Hyperarithmetic Theory . . . . . . . . . . . . . . . . . . . . 154
7.1.6 Subsystems of second order arithmetic . . . . . . . . . . . . 155
7.1.7 Linear orderings . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2 Between ACA and ∆1-CA . . . . . . . . . . . . . . . . . . . . . . . 157
0 1 0
7.3 Models of INDEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.1 The construction . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3.2 Pairs of computable structures. . . . . . . . . . . . . . . . . 167
7.4 The Game Statements . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.5 JI does not imply G1 . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.5.1 Ranked games . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.5.2 The forcing notion . . . . . . . . . . . . . . . . . . . . . . . 173
7.5.3 The forcing relation . . . . . . . . . . . . . . . . . . . . . . . 174
7.5.4 Retaggings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8 Ranked Structures and Arithmetic Transfinite Recursion (with
Noam Greenberg). 178
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.1.1 Reverse mathematics . . . . . . . . . . . . . . . . . . . . . . 179
8.1.2 The classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1.3 The statements . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.1.4 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.1.6 More Results . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.2.1 Equivalence over ACA . . . . . . . . . . . . . . . . . . . . . 186
0
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8.2.2 Proofs of arithmetic comprehension . . . . . . . . . . . . . . 188
8.3 Well-founded trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.3.1 Ranked Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.3.2 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.3.3 Proofs of arithmetic comprehension . . . . . . . . . . . . . . 194
8.4 Superatomic Boolean algebras . . . . . . . . . . . . . . . . . . . . . 195
8.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.4.2 Ranked Boolean algebras . . . . . . . . . . . . . . . . . . . . 199
8.4.3 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.4.4 Proofs of arithmetic comprehension . . . . . . . . . . . . . . 208
8.5 Reduced p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.5.1 Ranked p-groups . . . . . . . . . . . . . . . . . . . . . . . . 210
8.5.2 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.5.3 Proofs of arithmetic comprehension . . . . . . . . . . . . . . 214
8.6 Scattered and compact spaces . . . . . . . . . . . . . . . . . . . . . 215
8.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.6.2 Metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . 216
8.6.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.6.4 Ranked spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.6.5 Reversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.6.6 Proofs of arithmetic comprehension . . . . . . . . . . . . . . 223
8.7 Up to equimorphism, hyperarithmetic is recursive . . . . . . . . . . 225
III Computable Mathematics 227
9 Up to equimorphism, hyperarithmetic is recursive. 228
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9.2 General ideas of the Proof . . . . . . . . . . . . . . . . . . . . . . . 230
9.3 Signed Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.3.1 Natural sum of ordinals and ranks . . . . . . . . . . . . . . . 236
9.3.2 Signed forests and signed sequences . . . . . . . . . . . . . . 238
9.3.3 The complements . . . . . . . . . . . . . . . . . . . . . . . . 242
9.4 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10 Boolean Algebras, Tarski Invariants, and Index Sets (with Barbara
F. Csima and Richard A. Shore). 248
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
10.2 Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . 251
10.3 Counting Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.4 Dense Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . 256
10.5 Back-and-Forth relations . . . . . . . . . . . . . . . . . . . . . . . . 258
10.6 The Σ and the Π cases (Theorem 10.2.4) . . . . . . . . . . . . . . 260
n n
10.7 The Σ ∧Π cases (Theorem 10.2.6) . . . . . . . . . . . . . . . . . 263
n n
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