Overview

Undergraduate students with no prior instruction in mathematical logic will benefit from this multi-part text. Part I offers an elementary but thorough overview of mathematical logic of 1st order. Part II introduces some of the newer ideas and the more profound results of logical research in the 20th century. 1967 edition.
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Mathematical Logic

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Overview

Undergraduate students with no prior instruction in mathematical logic will benefit from this multi-part text. Part I offers an elementary but thorough overview of mathematical logic of 1st order. Part II introduces some of the newer ideas and the more profound results of logical research in the 20th century. 1967 edition.
Read More Show Less

Product Details

  • ISBN-13: 9780486317076
  • Publisher: Dover Publications
  • Publication date: 4/22/2013
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 416
  • Sales rank: 872,137
  • File size: 17 MB
  • Note: This product may take a few minutes to download.

Table of Contents

Part I. Elementary Mathematical Logic
Chapter I. The propositional calculus 3
1. Linguistic considerations: formulas 3
2. Model theory: truth tables, validity 8
3. Model theory: the substitution rule, a collection of valid formulas 13
4. Model theory: implication and equivalence 17
5. Model theory: chains of equivalences 20
6. Model theory: duality 22
7. Model theory: valid consequence 25
8. Model theory: condensed truth tables 28
9. Proof theory: provability and deducibility 33
10. Proof theory: the deduction theorem 39
11. Proof theory: consistency, introduction and elimination rules 43
12. Proof theory: completeness 45
13. Proof theory: use of derived rules 50
14. Applications to ordinary language: analysis of arguments 58
15. Applications to ordinary language: incompletely stated arguments 67
Chapter II. The predicate calculus 74
16. Linguistic considerations: formulas, free and bound occurrences of variables 74
17. Model theory: domains, validity 83
18. Model theory: basic results on validity 93
19. Model theory: further results on validity 96
20. Model theory: valid consequence 101
21. Proof theory: provability and deducibility 107
22. Proof theory: the deduction theorem 112
23. Proof theory: consistency, introduction and elimination rules 116
24. Proof theory: replacement, chains of equivalences 121
25. Proof theory: alterations of quantifiers, prenex form 125
26. Applications to ordinary language: sets, Aristotelian categorical forms 134
27. Applications to ordinary language: more on translating words into symbols 140
Chapter III. The predicate calculus with equality 148
28. Functions, terms 148
29. Equality 151
30. Equality vs. equivalence, extensionality 157
31. Descriptions 167
Part II. Mathematical Logic and the Foundations of Mathematics
Chapter IV. The foundations of mathematics 175
32. Countable sets 175
33. Cantor's diagonal method 180
34. Abstract sets 183
35. The paradoxes 186
36. Axiomatic thinking vs. intuitive thinking in mathematics 191
37. Formal systems, metamathematics 198
38. Formal number theory 201
39. Some other formal systems 215
Chapter V. Computability and decidability 223
40. Decision and computation procedures 223
41. Turing machines, Church's thesis 232
42. Church's theorem (via Turing machines) 242
43. Applications to formal number theory: undecidability (Church) and incompleteness (Godel's theorem) 247
44. Applications to formal number theory: consistency proofs (Godel's second theorem) 254
45. Application to the predicate calculus (Church, Turing) 260
46. Degrees of unsolvability (Post), hierarchies (Kleene, Mostowski) 265
47. Undecidability and incompleteness using only simple consistency (Rosser) 273
Chapter VI. The predicate calculus (additional topics) 283
48. Godel's completeness theorem: introduction 283
49. Godel's completeness theorem: the basic discovery 295
50. Godel's completeness theorem with a Gentzen-type formal system, the Lowenheim-Skolem theorem 305
51. Godel's completeness theorem (with a Hilbert-type formal system) 312
52. Godel's completeness theorem, and the Lowenheim-Skolem theorem, in the predicate calculus with equality 315
53. Skolem's paradox and nonstandard models of arithmetic 321
54. Gentzen's theorem 331
55. Permutability, Herbrand's theorem 338
56. Craig's interpolation theorem 349
57. Beth's theorem on definability, Robinson's consistency theorem 361
Bibliography 371
Theorem and lemma numbers: pages 386
List of postulates 387
Symbols and notations 388
Index 389
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