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Computable Structures and the Hyperarithmetical Hierarchy,9780444500724
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Computable Structures and the Hyperarithmetical Hierarchy


Author(s): Ash† ; Knight
ISBN10:  0444500723
ISBN13:  9780444500724
Format:  Hardcover
Pub. Date:  6/16/2000
Publisher(s): Elsevier Science & Technology

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SummaryTable of Contents
This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties).
Preface v
Introduction
v
Constructivizations
v
Authorship and acknowledgements
viii
Computability
1(20)
Informal concepts
1(3)
Church's Thesis
4(1)
Proof by Church's Thesis
5(1)
Some basic results
5(2)
Coding functions
7(1)
Kleene's Normal Form Theorem
8(1)
Facts about c.e. sets and relations
8(4)
The standard list of c.e. sets
12(1)
The s-m-n Theorem
12(1)
The Recursion Theorem
13(1)
Relative computability
14(2)
The relativized s-m-n Theorem
16(1)
Turing reducibility
16(1)
Other reducibilities
17(4)
The arithmetical hierarchy
21(12)
Jumps
21(1)
Basic definitions
22(2)
Basic theorems
24(2)
Combining arithmetical relations
26(1)
Alternative definitions
27(1)
Approximations
28(2)
Trees
30(3)
Languages and structures
33(24)
Propositional languages and structures
33(1)
Predicate languages
34(2)
Structures for a predicate language
36(1)
Satisfaction
37(1)
Enlarging a structure
38(1)
Theories and models
39(1)
Prenex normal form
40(1)
Isomorphism
40(1)
Quotient structures
41(1)
Model existence
42(2)
Compactness
44(1)
Some special kinds of structures
45(4)
Computable sets of sentences
49(1)
Complexity of structures
50(1)
Complexity of definable relations
51(2)
Copies of a given structure
53(2)
Complexity of quotient structures
55(2)
Ordinals
57(14)
Set theoretic facts
57(1)
Inductive proofs and definitions
58(1)
Operations on ordinals
59(1)
Cantor normal form
60(1)
Constructive and computable ordinals
60(1)
Kleene's O
61(1)
Constructive and computable ordinals
62(4)
Transfinite induction on ordinal notation
66(5)
The hyperarithmetical hierarchy
71(18)
The hyperarithmetical hierarchy
71(4)
The analytical hierarchy
75(7)
The Kleene-Brouwer ordering
82(2)
Relativizing
84(2)
Ershov's hierarchy
86(3)
Infinitary formulas
89(16)
Predicate formulas
89(1)
Sample formulas
90(2)
Subformulas and free variables
92(1)
Normal form
93(1)
Model existence
94(2)
Scott's Isomorphism Theorem
96(2)
Ranks and special Scott families
98(2)
Rigid structures and defining families
100(1)
Definability of relations
101(1)
Propositional formulas
102(3)
Computable infinitary formulas
105(16)
Informal definitions
105(2)
Formal definition
107(1)
Sample formulas
108(1)
Satisfaction and the hyperarithmetical hierarchy
109(1)
Further hierarchies of computable formulas
110(2)
Computable propositional formulas
112(3)
The simplest language
115(1)
Hyperarithmetical formulas
116(2)
X-computable formulas
118(3)
The Barwise-Kreisel Compactness Theorem
121(16)
Model existence and paths through trees
121(2)
The Compactness Theorem
123(4)
Hyperarithmetical saturation
127(2)
Orderings and trees
129(2)
Boolean algebras
131(1)
Groups
132(1)
Priority constructions
133(3)
Ranks
136(1)
Existence of computable structures
137(22)
Equivalence structures
137(2)
Abelian p-groups
139(3)
Linear orderings
142(8)
Boolean algebras
150(2)
Results of Wehner
152(3)
Decidable homogeneous structures
155(4)
Completeness and forcing
159(22)
Images of a relation
159(8)
Ershov's hierarchy
167(2)
Images of a pair of relations
169(4)
Isomorphisms
173(4)
Expansions
177(1)
Sets computable in all copies
177(2)
Copies of an arbitrary structure
179(2)
The Ash-Nerode Theorem
181(16)
Simple examples
181(1)
Results of Ash and Nerode
182(3)
Applications
185(2)
Vector spaces
185(1)
Algebraically closed fields
186(1)
Expansions
187(1)
Results of Harizanov
188(1)
Ershov's hierarchy
189(3)
Pairs of relations
192(5)
Computable categoricity and stability
197(16)
Simple examples
197(1)
Relations between notions
198(1)
Computable categoricity
199(2)
Decidable structures
201(2)
Computable stability
203(2)
Computable dimension
205(1)
One or infinitely many
206(3)
Quotient structures
209(4)
n-systems
213(14)
Introduction
213(1)
Statement of the metatheorem
214(2)
Some examples
216(4)
Proof of the metatheorem
220(3)
Looking ahead
223(1)
Michalski's Theorem
224(3)
α-systems
227(12)
Statement of the metatheorem
227(1)
Organizing the construction
228(2)
Derived systems
230(4)
Simultaneous runs
234(1)
Special (αn)-systems
235(2)
Further metatheorems
237(2)
Back-and-forth relations
239(14)
Standard back-and-forth relations
239(2)
α-friendly families
241(1)
Examples
242(5)
Arithmetic
242(1)
Vector spaces
242(2)
Linear orderings
244(2)
Boolean algebras
246(1)
Ranks
247(3)
Stronger back-and-forth relations
250(1)
Abstract back-and-forth relations
251(1)
Open problems
252(1)
Theorems of Barker and Davey
253(10)
Barker's Theorems
253(6)
Davey's Theorems
259(4)
Δ0α stability and categoricity
263(12)
Relations between notions
263(1)
Δ0α stability
264(4)
Well orderings
268(1)
Δ0α categoricity
269(3)
Superatomic Boolean algebras
272(3)
Pairs of computable structures
275(16)
Simple examples
275(3)
General results
278(3)
Results of Feiner and Thurber
281(3)
Limit structures
284(3)
Quotient orderings
287(4)
Models of arithmetic
291(20)
Scott sets
291(3)
Enumerations
294(2)
Structures representing a Scott set
296(9)
Harrington's Theorem
305(2)
Solovay's Theorems
307(1)
Sets computable in all models
308(1)
Open problems
309(2)
A Special classes of structures 311(12)
Vector spaces
311(1)
Fields
312(2)
Orderings
314(1)
Operations on orderings
314(1)
Boolean algebras
315(2)
Equivalence structures
317(1)
Abelian p-groups
317(1)
Models of arithmetic
318(5)
Matijasevic's Theorem
319(1)
Definability of satisfaction
320(1)
Independence
321(2)
Bibliography 323(12)
Index 335

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