Once we have accepted a precise replacement of the concept of algo rithm, it becomes possible to attempt the problem whether there exist well-defined collections of problems which cannot be handled by algo rithms, and if that is the case, to give concrete cases of this kind. Many such investigations were carried out during the last few decades. The undecidability of arithmetic and other mathematical theories was shown, further the unsolvability of the word problem of group theory. Many mathematicians consider these results and the theory on which they are based to be the most characteristic achievements of mathe matics in the first half of the twentieth century. If we grant the legitimacy of the suggested precise replacements of the concept of algorithm and related concepts, then we can say that the mathematicians have shown by strictly mathematical methods that there exist mathematical problems which cannot be dealt with by the methods of calculating mathematics. In view of the important role which mathematics plays today in our conception of the world this fact is of great philosophical interest. Post speaks of a natural law about the "limitations of the mathematicizing power of Homo Sapiens". Here we also find a starting point for the discussion of the question, what the actual creative activity of the mathematician consists in. In this book we shall give an introduction to the theory of algorithms.
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ISBN-13
978-3-540-04501-4 (9783540045014)
DOI
10.1007/978-3-642-46178-1
Schweitzer Klassifikation
1. Introductory Reflections on Algorithms.- § 1. The Concept of Algorithm.- § 2. The Fundamental Concepts of the Theory of Constructivity.- § 3. The Concept of Turing Machine as an Exact Mathematical Substitute for the Concept of Algorithm.- § 4. Historical Remarks.- 2. Turing Machines.- § 5. Definition of Turing Machines.- § 6. Precise Definition of Constructive Concepts by means of Turing Machines.- § 7. Combination of Turing Machines.- § 8. Special Turing Machines.- § 9. Examples of Turing-Computability and Turing-Decidability.- 3. ?-Recursive Functions.- § 10. Primitive Recursive Functions.- § 11. Primitive Recursive Predicates.- § 12. The ?-Operator.- § 13. Example of a Computable Function which is not Primitive Recursive.- § 14. ?-Recursive Functions and Predicates.- 4. The Equivalence of Turing-Computability and ?-Recursiveness.- § 15. Survey. Standard Turing-Computability.- § 16. The Turing-Computability of ?-Recursive Functions.- § 17. Gödel Numbering of Turing Machines.- § 18. The ?-Recursiveness of Turing-Computable Functions. Kleene's Normal Form.- 5. Recursive Functions.- § 19. Definition of Recursive Functions.- § 20. The Recursiveness of ?-Recursive Functions.- § 21. The ?-Recursiveness of Recursive Functions.- 6. Undecidable Predicates.- § 22. Simple Undecidable Predicates.- § 23. The Unsolvability of the Word Problem for Semi-Thue Systems and Thue Systems.- § 24. The Predicate Calculus.- § 25. The Undecidability of the Predicate Calculus.- § 26. The Incompleteness of the Predicate Calculus of the Second Order.- § 27. The Undecidability and Incompleteness of Arithmetic.- 7. Miscellaneous.- § 28. Enumerable Predicates.- § 29. Arithmetical Predicates.- § 30. Universal Turing Machines.- § 31. ?-K-Definability.- § 32. The Minimal Logic of Fitch.- § 33. Further Precise Mathematical Replacements of the Concept of Algorithm.- § 34. Recursive Analysis.- Author and Subject Index.
