Truth, Proof and Infinity: A Theory of Constructive Reasoning

1. Introduction and Statement of the Problem.- 2. What's Wrong with Set Theory?.- 3. What's Wrong with Infinite Quantifiers?.- 4. Abstraction and Idealisation.- 5. What are Constructions?.- 6. Truth and Proof of Logical Formulae.- 7. The Need for a Theory of Constructions.- 8. Theories of Constructions.- 9. Hilbert's Formalism.- 10. Open-endedness.- 11. Analysis.- 12. Introduction to Part II.- 13. Design of the Term Language.- 14. The Term Language.- 15. From the Term Language to the Expanded Term Language.- 16. The Expanded Term Language.- 17. The Protological Sequent Calculus.- 18. Commentary on the Protological Axioms and Rules.- 19. From Protologic to Expanded Protologic.- 20. Expanded Protologic.- 21. From Expanded Protologic to the Coding of Trees.- 22. The Coding of Trees.- 23. The Expanded Term Language as a Functional Programming Language.- 24. Introduction to Part III.- 25. From the Coding of Trees to Logic.- 26. Logic.- 27. From Logic to the Calculus of Proof Functions.- 28. Calculus of Proof Functions.- 29. From Calculus of Proof Functions to the Logic of Partial Terms.- 30. Logic of Partial Terms.- 31. From Logic of Partial Terms to Heyting Arithmetic.- 32. Heyting Arithmetic.- 33. From Heyting Arithmetic to Peano Arithmetic.- 34. Peano Arithmetic.- 35. Conclusions on Arithmetic.- 36. Introduction to Part IV.- 37. From Expanded Protologic to the Second-Order Coding of Trees.- 38. The Second-Order Coding of Trees.- 39. From the Second-Order Coding of Trees to Second-Order Logic.- 40. Second-Order Logic.- 41. From Second-Order Logic to Second-Order Calculus of Proof Functions.- 42. Second-Order Calculus of Proof Functions.- 43. From Second-Order Calculus of Proof Functions to Second-Order Logic of Partial Terms.- 44. Second-Order Logic of Partial Terms.- 45. From Second-Order Logic of Partial Terms to Second-Order Heyting Arithmetic.- 46. Second-Order Heyting Arithmetic.- 47. From Second-Order Heyting Arithmetic to Second-Order Peano Arithmetic.- 48. Second-Order Peano Arithmetic.- 49. Conclusions on Analysis.- References.- Index of symbols.- Index of axioms, theorems and rules of inference.- Index of names.- Index of topics.