Mathematical Logic

di H.-D. Ebbinghaus (Autore)

J. Flum (Autore)

Wolfgang Thomas (Autore)

Springer-Verlag New York Inc., 1994

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.

• Autore: H.-D. Ebbinghaus, J. Flum, Wolfgang Thomas
• Editore: Springer-Verlag New York Inc.
• Collana: Undergraduate Texts in Mathematics
• Anno: 1994
• Rilegatura: Hardback
• Pagine: 291 p.
• Dimensioni: 235 x 155 mm
• EAN: 9780387942582

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