The Structure of Models of Peano Arithmetic

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ISBN: 9780198568278 | 0198568274
Cover: Hardcover
Copyright: 9/14/2006

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Aimed at research logicians and mathematicians, this much-awaited monograph covers over forty years of work on relative classification theory for non-standard models of arithmetic. With graded exercises at the end of each chapter, the book covers basic isomorphism invariants: families of types realized in a model, lattices of elementary substructures and automorphism groups. Many results involve applications of the powerful technique of minimal types due to Haim Gaifman, and some of the results are classical but have never been published in a book form before.

Basics	p. 1
Notation and basic definitions	p. 1
Skolem closures	p. 4
End extensions and cofinal extensions	p. 5
Coding bounded sets and classes	p. 6
Standard systems	p. 8
Types	p. 9
Blass-Gaifman and Ehrenfeucht lemmas	p. 10
Recursive saturation and arithmetic saturation	p. 12
Satisfaction classes and resplendency	p. 14
Cuts and gaps in recursively saturated models	p. 16
Truth definitions and restricted saturation	p. 18
Arithmetized Completeness Theorem	p. 19
Friedman's Embedding Theorem	p. 20
Exercises	p. 21
Remarks & References	p. 22
Extensions	p. 25
Simple extensions	p. 25
Minimal extensions	p. 27
Superminimal extensions	p. 29
Greatest common initial segments	p. 31
The MacDowell-Specker Theorem	p. 33
Superminimal conservative extensions	p. 37
Rather classless models	p. 37
Ramsey's Theorem in ACA₀	p. 38
Amalgamations	p. 39
Nonelementary extensions	p. 43
Exercises	p. 45
Remarks & References	p. 47
Minimal and Other Types	p. 49
Types related to indiscernibility	p. 49
Indiscernible types	p. 49
n-indiscernible types	p. 53
End-extensional types	p. 55
Rare types	p. 56
Minimal types	p. 59
Selective types	p. 59
Characterizing minimal types	p. 61
An example	p. 65
Canonical extensions	p. 66
Products of types	p. 67
The automorphism group	p. 69
The substructure lattice	p. 72
Resolute types	p. 73
The Paris-Mills theorems	p. 78
Exercises	p. 84
Remarks & References	p. 88
Substructure Lattices	p. 89
Lattices	p. 89
Substructure lattices	p. 94
Finite distributive lattices, I	p. 97
Finite distributive lattices, II	p. 102
Finite lattices	p. 105
The pentagon lattice	p. 122
Infinite distributive lattices	p. 127
Exercises	p. 130
Remarks & References	p. 132
How to Control Types	p. 135
Solid bases and AH-sets	p. 135
Controlling indiscernibles and automorphisms	p. 136
AH-sets	p. 137
The proof	p. 139
True arithmetic	p. 142
Omitting indiscernibles	p. 144
Hanf numbers	p. 145
The automorphism group	p. 149
Indiscernible generators	p. 151
Exercises	p. 156
Remarks & References	p. 157
Generics and Forcing	p. 158
Generics	p. 158
Forcing	p. 160
Definition	p. 160
n-Generics	p. 161
Prime expansions	p. 162
The Low Basis Theorem	p. 163
Product forcing	p. 165
MacDowell-Specker vs the uncountable	p. 167
No end extension	p. 168
Extensions with mutual generics	p. 169
Getting many classes	p. 172
Perfect generics	p. 175
Exercises	p. 178
Remarks & References	p. 179
Cuts	p. 180
Semiregular cuts	p. 181
Semiregularity and WKL₀	p. 183
Regular cuts	p. 184
Many faces of strongness	p. 189
Why PA?	p. 194
Schemes axiomatizing arithmetic	p. 196
Exercises	p. 200
Remarks & References	p. 201
Automorphisms of Recursively Saturated Models	p. 202
Moving undefinable elements	p. 202
Moving cuts and classes	p. 203
Moving gaps	p. 205
Back-and-forth	p. 206
Extending automorphisms	p. 210
Maximal automorphisms	p. 214
Fixing strong cuts	p. 217
Topology on the automorphism group	p. 219
Maximal point stabilizers	p. 221
Arithmetic saturation and open subgroups	p. 223
Exercises	p. 225
Remarks & References	p. 227
Automorphism Groups of Recursively Saturated Models	p. 230
Generic automorphisms	p. 230
Dense conjugacy classes	p. 235
Small index property	p. 239
The confinality of the automorphism group	p. 242
Property FA	p. 244
Coding the standard system	p. 245
The spectrum of automorphism groups	p. 251
Exercises	p. 253
Remarks & References	p. 254
¿₁-like Models	p. 256
w1-Like recursively saturated models	p. 256
Similar nonisomorphic models	p. 260
Finitely determinate structures and PA(aa)	p. 265
Ramsey quantifiers and PA(Q²)	p. 269
Rigid recursively saturated models	p. 273
Isomorphic + nonisomorphic x	p. 275
Exercises	p. 276
Remarks & References	p. 278
Order Types	p. 281
On (k, k)-cuts	p. 281
Saturation of the order reduct	p. 282
Canonical codes	p. 283
N₁-saturation	p. 283
k-saturation for k > N₁	p. 284
Exercises	p. 287
Remarks & References	p. 288
Twenty Questions	p. 289
References	p. 295
Index of Names	p. 307
Index of Terms	p. 309
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The Structure of Models of Peano Arithmetic

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