PART I. ELEMENTARY MATHEMATICAL LOGIC
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CHAPTER I. THE PROPOSITIONAL CALCULUS |
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1. Linguistic considerations: formulas |
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2. "Model theory: truth tables,validity " |
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3. "Model theory: the substitution rule, a collection of valid formulas" |
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4. Model theory: implication and equivalence |
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5. Model theory: chains of equivalences |
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6. Model theory: duality |
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7. Model theory: valid consequence |
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8. Model theory: condensed truth tables |
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9. Proof theory: provability and deducibility |
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10. Proof theory: the deduction theorem |
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11. "Proof theory: consistency, introduction and elimination rules" |
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12. Proof theory: completeness |
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13. Proof theory: use of derived rules |
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14. Applications to ordinary language: analysis of arguments |
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15. Applications to ordinary language: incompletely stated arguments |
CHAPTER II. THE PREDICATE CALCULUS |
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16. "Linguistic considerations: formulas, free and bound occurrences of variables" |
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17. "Model theory: domains, validity" |
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18. Model theory: basic results on validity |
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19. Model theory: further results on validity |
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20. Model theory: valid consequence |
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21. Proof theory: provability and deducibility |
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22. Proof theory: the deduction theorem |
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23. "Proof theory: consistency, introduction and elimination rules" |
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24. "Proof theory: replacement, chains of equivalences" |
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25. "Proof theory: alterations of quantifiers, prenex form" |
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26. "Applications to ordinary language: sets, Aristotelian categorical forms" |
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27. Applications to ordinary language: more on translating words into symbols |
CHAPTER III. THE PREDICATE CALCULUS WITH EQUALITY |
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28. "Functions, terms" |
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29. Equality |
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30. "Equality vs. equivalence, extensionality" |
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31. Descriptions |
PART II. MATHEMATICAL LOGIC AND THE FOUNDATIONS OF MATHEMATICS |
CHAPTER IV. THE FOUNDATIONS OF MATHEMATICS |
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32. Countable sets |
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33. Cantor's diagonal method |
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34. Abstract sets |
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35. The paradoxes |
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36. Axiomatic thinking vs. intuitive thinking in mathematics |
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37. "Formal systems, metamathematics" |
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38. Formal number theory |
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39. Some other formal systems |
CHAPTER V. COMPUTABILITY AND DECIDABILITY |
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40. Decision and computation procedures |
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41. "Turing machines, Church's thesis" |
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42. Church's theorem (via Turing machines) |
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43. Applications to formal number theory: undecidability (Church) and incompleteness (Gödel's theorem) |
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44. Applications to formal number theory: consistency proofs (Gödel's second theorem) |
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45. "Application to the predicate calculus (Church, Turing)" |
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46. "Degrees of unsolvability (Post), hierarchies (Kleene, Mostowski)." |
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47. Undecidability and incompleteness using only simple consistency (Rosser) |
CHAPTER VI. THE PREDICATE CALCULUS (ADDITIONAL TOPICS) |
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48. Gödel's completeness theorem: introduction |
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49. Gödel's completeness theorem: the basic discovery |
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50. "Gödel's completeness theorem with a Gentzen-type formal system, the Löwenheim-Skolem theorem" |
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51. Gödel's completeness theorem (with a Hilbert-type formal system) |
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52. "Gödel's completeness theorem, and the Löwenheim-Skolem theorem, in the predicate calculus with equality" |
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53. Skolen's paradox and nonstandard models of arithmetic |
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54. Gentzen's theorem |
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55. "Permutability, Herbrand's theorem" |
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56. Craig's interpolation theorem |
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57. "Beth's theorem on definability, Robinson's consistency theorem" |
BIBLIOGRAPHY |
THEOREM AND LEMMA NUMBERS: PAGES |
LIST OF POSTULATES |
SYMBOLS AND NOTATIONS |
INDEX |
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