Fundamentals of mathematical logic

895 Pages·2005·12.123 MB·English

by  Hinman, Peter G.

Fundamentals of
Mathematical Logic

Fundamentals of
Mathematical Logic
Peter G. Hinman
University of Michigan
A K Peters
Wellesley, Massachusetts
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Library of Congress Cataloging-in-Publication Data
Hinman,PeterG.
Fundamentalsofmathematicallogic/PeterG.Hinman.
p. cm.
Includesbibliographicalreferencesandindexes.
ISBN1-56881-262-0
1. Logic, Symbolicandmathematical. 2. Modeltheory. 3. Settheory. 4. Recursion
theory. I.Title.
QA9.H5272005
511.3--dc22
2005050968
Printed in India
09 08 07 06 05 10 9 8 7 6 5 4 3 2 1
for Annika Michele and Celia Katherine

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Propositional Logic and Other Fundamentals . . . . 13
1.1. The propositional language . . . . . . . . . . . . . . . . 13
1.2. Induction and recursion . . . . . . . . . . . . . . . . . 20
Induction . . . . . . . . . . . . . . . . . . . . . . . 20
Recursion . . . . . . . . . . . . . . . . . . . . . . . 25
1.3. Propositionalsemantics . . . . . . . . . . . . . . . . . 32
1.4. Propositionaltheories . . . . . . . . . . . . . . . . . . 41
General properties . . . . . . . . . . . . . . . . . . . 42
Compactness . . . . . . . . . . . . . . . . . . . . . . 47
1.5. Decidability and effective enumerability . . . . . . . . . . 54
1.6. Other constructions . . . . . . . . . . . . . . . . . . . 63
Notions of consistency . . . . . . . . . . . . . . . . . . 63
Ultraproducts . . . . . . . . . . . . . . . . . . . . . 67
1.7. Topology and Boolean algebra . . . . . . . . . . . . . . 72
Topology . . . . . . . . . . . . . . . . . . . . . . . 73
Boolean algebra . . . . . . . . . . . . . . . . . . . . 74
viii Contents
2. First-Order Logic . . . . . . . . . . . . . . . . . . . 83
2.1. Syntax and semantics of first-order languages . . . . . . . . 83
2.2. Basic semantics . . . . . . . . . . . . . . . . . . . . . 96
Substitution . . . . . . . . . . . . . . . . . . . . . . 105
2.3. Structures . . . . . . . . . . . . . . . . . . . . . . . 114
Isomorphism and equivalence . . . . . . . . . . . . . . . 115
Substructures . . . . . . . . . . . . . . . . . . . . . 119
Products and chains . . . . . . . . . . . . . . . . . . . 130
2.4. Theories . . . . . . . . . . . . . . . . . . . . . . . . 139
The language of equality . . . . . . . . . . . . . . . . . 149
Dense linear orderings . . . . . . . . . . . . . . . . . . 154
2.5. Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 160
2.6. Changing languages . . . . . . . . . . . . . . . . . . . 173
Interpretations . . . . . . . . . . . . . . . . . . . . . 186
3. Completeness and Compactness . . . . . . . . . . . 193
3.1. Countable compactness . . . . . . . . . . . . . . . . . 194
3.2. Countable completeness . . . . . . . . . . . . . . . . . 204
3.3. Other constructions . . . . . . . . . . . . . . . . . . . 216
Notions of consistency . . . . . . . . . . . . . . . . . . 216
Ultraproducts . . . . . . . . . . . . . . . . . . . . . 224
Boolean algebra . . . . . . . . . . . . . . . . . . . . 228
3.4. Uncountable languages and structures . . . . . . . . . . . 236
3.5. Applications of compactness . . . . . . . . . . . . . . . 249
3.6. Higher-order logic . . . . . . . . . . . . . . . . . . . . 276
Monadic second-order logic . . . . . . . . . . . . . . . . 276
3.7. Infinitary logic . . . . . . . . . . . . . . . . . . . . . 293
4. Incompleteness and Undecidability . . . . . . . . . 309
4.1. A first look . . . . . . . . . . . . . . . . . . . . . . 310
4.2. Recursive functions and relations . . . . . . . . . . . . . 326
4.3. Recursively enumerable sets and relations . . . . . . . . . 341
4.4. G¨odel numbering . . . . . . . . . . . . . . . . . . . . 352
4.5. Definability in arithmetic I . . . . . . . . . . . . . . . . 364
4.6. Representability: First Incompleteness Theorem . . . . . . 369
5. Topics in Definability . . . . . . . . . . . . . . . . . 393
5.1. Definability in arithmetic II . . . . . . . . . . . . . . . 393
5.2. Indexing . . . . . . . . . . . . . . . . . . . . . . . . 409
5.3. Second Incompleteness Theorem . . . . . . . . . . . . . 421
Contents ix
5.4. Church’s Thesis . . . . . . . . . . . . . . . . . . . . 431
Recursion equations . . . . . . . . . . . . . . . . . . . 432
Abstract machines . . . . . . . . . . . . . . . . . . . 436
5.5. Applications to other languages and theories . . . . . . . . 443
6. Set Theory . . . . . . . . . . . . . . . . . . . . . . . 455
6.1. Zermelo-Fraenkelset theory . . . . . . . . . . . . . . . 456
6.2. Mathematics in set theory I . . . . . . . . . . . . . . . 472
6.3. Ordinal numbers: induction and recursion . . . . . . . . . 497
6.4. Cardinal numbers . . . . . . . . . . . . . . . . . . . . 510
6.5. Models and independence . . . . . . . . . . . . . . . . 527
6.6. Mathematics in set theory II . . . . . . . . . . . . . . . 550
6.7. The constructible universe . . . . . . . . . . . . . . . . 567
6.8. Generic extensions . . . . . . . . . . . . . . . . . . . 577
6.9. Forcing . . . . . . . . . . . . . . . . . . . . . . . . 596
6.10. Large cardinals . . . . . . . . . . . . . . . . . . . . 605
6.11. Determinacy . . . . . . . . . . . . . . . . . . . . . 622
7. Model Theory . . . . . . . . . . . . . . . . . . . . . 655
7.1. Partialembeddings . . . . . . . . . . . . . . . . . . . 655
7.2. Boolean algebras, ultrafilters and types . . . . . . . . . . 671
7.3. Countable models of countable theories . . . . . . . . . . 683
7.4. Uncountable models of countable theories . . . . . . . . . 700
7.5. Morley’s Theorem . . . . . . . . . . . . . . . . . . . 708
7.6. Abstract logics . . . . . . . . . . . . . . . . . . . . . 721
8. Recursion Theory . . . . . . . . . . . . . . . . . . . 733
8.1. Many-one degrees and r.e. sets . . . . . . . . . . . . . . 733
8.2. Turing reducibility . . . . . . . . . . . . . . . . . . . 756
8.3. The jump operator . . . . . . . . . . . . . . . . . . . 770
8.4. Upper bounds . . . . . . . . . . . . . . . . . . . . . 783
8.5. Jumps of r.e. sets . . . . . . . . . . . . . . . . . . . . 793
8.6. Lower bounds . . . . . . . . . . . . . . . . . . . . . 808
References . . . . . . . . . . . . . . . . . . . . . . . . 821
Item References . . . . . . . . . . . . . . . . . . . . . 829
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . 835
Subject Index . . . . . . . . . . . . . . . . . . . . . . . 857