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CHAPTER 0．PREREQUISITES 1

CHAPTER 1．BEGINNING MATHEMATICAL LOGIC 5

1．General considerations 5

2．Structures and formal languages 9

3．Higher-order languages 14

4．Basic syntax 15

5．Notational conventions 18

6．Propositional semantics 20

7．Propositional tableaux 25

8．The Elimination Theorem for propositional tableaux 31

9．Completeness of propositional tableaux 33

10．The propositional calculus 34

11．The propositional calculus and tableaux 40

12．Weak completeness of the propositional calculus 43

13．Strong completeness of the propositional calculus 44

14．Propositionallogic based on ？ and∧ 46

15．Propositional logic based on？,→,∧and ∨ 47

16．Historical and bibliographical remarks 48

CHAPTER 2．FIRST-ORDER LOGIC 49

1．First-order semantics 49

2．Freedom and bondage 54

3．Substitution 57

4．First-order tableaux 67

5．Some"book-keeping"lemmas 72

6．The Elimination Theorem for first-order tableaux 79

7．Hintikka sets 83

8．Completeness of first-order tableaux 88

9．Prenex and Skolem forms 93

10．Elimination of function symbols 97

11．Elimination of equality 101

12．Relativization 102

13．Virtual terms 104

14．Historical and bibliographical remarks 107

CHAPTER 3．FIRST-ORDER LOGIC(CONTINUED) 108

1．The first-order predicate calculus 108

2．The first-order predicate calculus and tableaux 115

3．Completeness of the first-order predicate calculus 117

4．First-order logic based on 3 122

5．What have we achieved? 122

6．Historical and bibliographical remarks 124

CHAPTER 4．BOOLEAN ALGEBRAS 125

1．Lattices 125

2．Boolean algebras 129

3．Filters and homomorphisms 133

4．The Stone Representation Theorem 141

5．Atoms 150

6．Duality for homomorphisms and continuous mappings 153

7．The Rasiowa-Sikorski Theorem 157

8．Historical and bibliographical remarks 159

CHAPTER 5．MODEL THEORY 161

1．Basic ideas of model theory 161

2．The L？wenheim-Skolem Theorems 168

3．Ultraproducts 174

4．Completeness and categoricity 184

5．Lindenbaum algebras 191

6．Element types and ？-categoricity 203

7．Indiscernibles and models with automorphisms 214

8．Historical and bibliographical remarks 224

CHAPTER 6．RECURSION THEORY 226

1．Basic notation and terminology 226

2．Algorithmic functions and functionals 230

3．The computer URIM 232

4．Computable functionals and functions 237

5．Recursive functionals and functions 239

6．A stockpile of examples 247

7．Church's Thesis 257

8．Recursiveness of computable functionals 259

9．Functionals with several sequence arguments 265

10．Fundamental theorems 266

11．Recursively enumerable sets 277

12．Diophantine relations 284

13．The Fibonacci sequence 288

14．The power function 296

15．Bounded universal quantification 305

16．The MRDP Theorem and Hilbert's Tenth Problem 311

17．Historical and bibliographical remarks 314

CHAPTER 7．LOGIC—LIMITATIVE RESULTS 316

1．General notation and terminology 316

2．Nonstandard models of Ω 318

3．Arithmeticity 324

4．Tarski's Theorem 327

5．Axiomatic theories 332

6．Baby arithmetic 334

7．Junior arithmetic 336

8．A finitely axiomatized theory 340

9．First-order Peano arithmetic 342

10．Undecidability 347

11．Incompleteness 353

12．Historical and bibliographical remarks 359

CHAPTER 8．RECURSION THEORY(CONTINUED) 361

1．The arithmetical hierarchy 361

2．A result concerning TΩ 369

3．Encoded theories 370

4．Inseparable pairs of sets 372

5．Productive and creative sets;reducibility 376

6．One-one reducibility;recursive isomorphism 384

7．Turing degrees 388

8．Post's problem and its solution 392

9．Historical and bibliographical remarks 398

CHAPTER 9．INTUITIONISTIC FIRST-ORDER LOGIC 400

1．Preliminary discussion 400

2．Philosophical remark 403

3．Constructive meaning of sentences 403

4．Constructive interpretations 404

5．Intuitionistic tableaux 408

6．Kripke's semantics 416

7．The Elimination Theorem for intuitionistic tableaux 422

8．Intuitionistic propositional calculus 433

9．Intuitionistic predicate calculus 434

10．Completeness 438

11．Translations from classical to intuitionistic logic 442

12．The Interpolation Theorem 445

13．Some results in classical logic 452

14．Historical and bibliographical remarks 457

CHAPTER 10．AXIOMATIC SET THEORY 459

1．Basic developments 459

2．Ordinals 468

3．The Axiom of Regularity 477

4．Cardinality and the Axiom of Choice 487

5．Reflection Principles 491

6．The formalization of satisfaction 497

7．Absoluteness 502

8．Constructible sets 509

9．The consistency of AC and GCH 516

10．Problems 522

11．Historical and bibliographical remarks 529

CHAPTER 11．NONSTANDARD ANALYSIS 531

1．Enlargements 532

2．Zermelo structures and their enlargements 536

3．Filters and monads 543

4．Topology 553

5．Topological groups 561

6．The real numbers 566

7．A methodological discussion 572

8．Historical and bibliographical remarks 573

BIBLIOGRAPHY 576

GENERAL INDFX 584

INDEX OF SYMBOLS 595