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Mathematics | Mathematical Logic

Mathematical Logic

Ebbinghaus, H.-D., Flum, J., Thomas, Wolfgang

2nd ed. 1994, X, 291 p.

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What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathe­ matical proofs? Only in this century has there been success in obtaining substantial and satisfactory answers. The present book contains a systematic discussion of these results. The investigations are centered around first-order logic. Our first goal is Godel's completeness theorem, which shows that the con­ sequence relation coincides with formal provability: By means of a calcu­ lus consisting of simple formal inference rules, one can obtain all conse­ quences of a given axiom system (and in particular, imitate all mathemat­ ical proofs). A short digression into model theory will help us to analyze the expres­ sive power of the first-order language, and it will turn out that there are certain deficiencies. For example, the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis. On the other hand, this difficulty can be overcome--even in the framework of first-order logic-by developing mathematics in set-theoretic terms. We explain the prerequisites from set theory necessary for this purpose and then treat the subtle relation between logic and set theory in a thorough manner.

Content Level » Lower undergraduate

Keywords » Arithmetic - Equivalence - Logic - Mathematische Logik - compactness theorem - mathematical logic - model theory - proof

Related subjects » Mathematics - Mathematics Education

Table of contents 

Preface; Part A: 1. Introduction; 2. Syntax of First-Order Languages; 3. Semantics of first-Order Languages; 4. A Sequent Calculus; 5. The Completeness Theorem; 6. The Lowenheim-Skolem and the Compactness Theorem; 7. The Scope of First-Order Logic; 8. Syntactic Interpretations and Normal Forms; Part B: 9. Extensions of First-Order Logic; 10. Limitations of the Formal Method; 11. Free Models and Logic Programming; 12. An Algebraic Characterization of Elementary Equivalence; 13. Lindstroem's Theorems; References; Symbol Index; Subject Index

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