Proof Theory

Preface......Page 3 1. Formalization of statements......Page 4 2. Formal proofs and related concepts......Page 8 3. A formulation of intuitionistic predicate calculus......Page 18 4. Axiom systems......Page 20 5. The cut-elimination theorem......Page 21 6. Some consequences of the cut-elimination theorem......Page 28 7. The predicate calculus with equality......Page 38 8. The completeness theorem......Page 41 9. A formulation of Peano arithmetic......Page 67 10. The incompleteness theorem......Page 72 11. A discussion of ordinals from a finitist standpoint......Page 80 12. A consistency proof of PA......Page 96 13. Provable well-orderings......Page 114 14. An additional topic......Page 123 15. Second order predicate calculus......Page 126 16. Some systems of second order predicate calculus......Page 134 17. The theory of relativization......Page 145 18. Truth definition for first order arithmetic......Page 150 19. The interpretation of a system of second order arithmetic......Page 156 20. Simple type theory......Page 161 21. The cut elimination theorem for simple type theory......Page 167 4. Infinitary Logic......Page 180 22. Infinitary logic with homogeneous quantifiers......Page 183 23. Determinate logic......Page 211 24. A general theory of heterogeneous quantifiers......Page 242 25. Introduction......Page 277 26. Ordinal diagrams......Page 282 27. A consistency proof of second order arithmetic with the Π^1_1-comprehension axiom......Page 309 28. A consistency proof for a system with inductive definitions......Page 325 29. Provable well-orderings......Page 337 30. The Π^1_1-comprehension axiom and the ω-rule......Page 339 31. Reflection principles......Page 345 Index......Page 358