In this post we will see Mathematical Logic by *Yu. L. Ershov,** E. A. Palyutin.*

This book presents in a systematic way a number of topics in modern

mathematical logic and the theory of algorithms. It can be used as

both a text book on mathematical logic for university students and

a text for specialist courses. The sections corresponding to the

obligatory syllabus (Sections 1 to 9 of Chapter 1,without the small

type, Sections 10 and 11 of Chapter 2, Sections 15 and 16 of Chapter

3,Sections 18 to 20, 22 and 23 of Chapter 4 and Section 35of Chapter 7) are written more thoroughly and in more detail than the sectionsrelating to more special questions.

The exposition of the propositional calculus and the calculus of predicates is not a conventional one, beginning as it does with a study of sequential variants of the calculi of natural deduction( although the traditional calculi, referred to as Hilbertian ,also appears here). The reasons for this are:

A) the possibility of providing a good explanation of the meaning of all the rules of inference;

B) the possibility of acquiring more rapidly the knack of making formal proofs;

C) a practical opportunity of making all the formal proofs necessary in the course for these calculi.

This book was translated from the Russian by *Vladimir Shokurov*. The book was published by first Mir Publishers in 1984.

All credits to the *original uploader.*

DJVU | OCR | 4.3 MB | Pages: 302 |

Update 31 May 2018: Added Internet Archive link

Table of Contents

Preface 7

**INTRODUCTION 9**

**Chapter 1.**

** THE PROPOSITIONAL CALCULUS 15**

1. Sets and words 15

2. The language of the propositional calculus 21

3. Axiom system and rules of inference 25

4. The equivalence of formulas 32

5. Normal forms35

6. Semantics of the propositional calculus 43

7. Characterization of provable formula 48

8. Hilbertian propositional calculus 52

9. Conservative extension of calculi 56

**Chapter 2.**

** SET THEORY 65**

10. Predicates and mappings 65

11. Partially ordered sets 70

12. Filters of Boolean algebra 78

13.The power of a set 82

14.The axiom of choice 90

**Chapter 3. **

** TRUTH ON ALGEBRAIC SYSTEMS 96**

15. Algebraic systems96

16. Formulas of the signature $\Sigma$ 102

17. Compactness theorem110

**Chapter 4.**

** THE CALCULUS OF PREDICATES 117**

18. Axioms and rules of inference 117

19. The equivalence of formulas 126

20. Normal forms 130

21. Theorem on the existence of a model 132

22. Hilbertian calculus of predicates 139

23. Pure calculus of predicates 144

**Chapter 5.**

** MODEL THEORY 149**

24. Elementary equivalence 149

25. Axiomatizable classes 157

26. Skolem functions 165

27. Mechanism of compatibility 168

28. Countable homogeneity and universality 181

29. Categoricity 188

**Chapter6.**

** PROOF THEORY 198**

30. The Gentzen system G 198

31. The invertibility of rules 204

32. Comparison of the calculi $CP^\Sigma$ and G 210

33. Herbrand theorem 217

34. The calculi of resolvents 228

**Chapter7.**

** ALGORITHMS AND RECURSIVE FUNCTIONS236**

35. Normal algorithms and Turing machines 236

36. Recursive functions 247

37. Recursively enumerable predicates 264

38. Undecidability of the calculus of predicates and Godel’s incompleteness theorem 276

List of symbols 292

Subject Index 295

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This book presents in a systematic way a number of topics in modern

mathematical logic and the theory of algorithms. It can be used as

both a text book on mathematical logic for university students and

a text for specialist courses. The sections corresponding to the

obligatory syllabus (Sections 1 to 9 of Chapter 1,without the small

type, Sections 10 and 11 of Chapter 2, Sections 15 and 16 of Chapter

3,Sections 18 to 20, 22 and 23 of Chapter 4 and Section 35of Chapter 7) are written more thoroughly and in more detail than the sectionsrelating to more special questions.

The exposition of the propositional calculus and the calculus of predicates is not a conventional one, beginning as it does with a study of sequential variants of the calculi of natural deduction( although the traditional calculi, referred to as Hilbertian ,also appears here). The reasons for this are:

A) the possibility of providing a good explanation of the meaning of all the rules of inference;

B) the possibility of acquiring more rapidly the knack of making formal proofs;

C) a practical opportunity of making all the formal proofs necessary in the course for these calculi.

This book was translated from the Russian by Vladimir Shokurov. The book was published by first Mir Publishers in 1984.

All credits to the original uploader.

DJVU | OCR | 4.3 MB | Pages: 302 |

You can get the book here

For magnet / torrent links go here.

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Table of Contents

Preface 7

INTRODUCTION 9

Chapter 1.

THE PROPOSITIONAL CALCULUS 15

1. Sets and words 15

2. The language of the propositional calculus 21

3. Axiom system and rules of inference 25

4. The equivalence of formulas 32

5. Normal forms35

6. Semantics of the propositional calculus 43

7. Characterization of provable formula 48

8. Hilbertian propositional calculus 52

9. Conservative extension of calculi 56

Chapter 2.

SET THEORY 65

10. Predicates and mappings 65

11. Partially ordered sets 70

12. Filters of Boolean algebra 78

13.The power of a set 82

14.The axiom of choice 90

Chapter 3.

TRUTH ON ALGEBRAIC SYSTEMS 96

15. Algebraic systems96

16. Formulas of the signature $\Sigma$ 102

17. Compactness theorem110

Chapter 4.

THE CALCULUS OF PREDICATES 117

18. Axioms and rules of inference 117

19. The equivalence of formulas 126

20. Normal forms 130

21. Theorem on the existence of a model 132

22. Hilbertian calculus of predicates 139

23. Pure calculus of predicates 144

Chapter 5.

MODEL THEORY 149

24. Elementary equivalence 149

25. Axiomatizable classes 157

26. Skolem functions 165

27. Mechanism of compatibility 168

28. Countable homogeneity and universality 181

29. Categoricity 188

Chapter6.

PROOF THEORY 198

30. The Gentzen system G 198

31. The invertibility of rules 204

32. Comparison of the calculi $CP^\Sigma$ and G 210

33. Herbrand theorem 217

34. The calculi of resolvents 228

Chapter7.

ALGORITHMS AND RECURSIVE FUNCTIONS236

35. Normal algorithms and Turing machines 236

36. Recursive functions 247

37. Recursively enumerable predicates 264

38. Undecidability of the calculus of predicates and Godel’s incompleteness theorem 276

List of symbols 292

Subject Index 295

thank u v m