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AuthorEbbinghaus, H.-D. author
TitleMathematical Logic [electronic resource] / by H.-D. Ebbinghaus, J. Flum, W. Thomas
ImprintNew York, NY : Springer New York : Imprint: Springer, 1994
Edition Second Edition
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Descript X, 291 p. online resource


What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out matheยญ 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 conยญ sequence relation coincides with formal provability: By means of a calcuยญ lus consisting of simple formal inference rules, one can obtain all conseยญ quences of a given axiom system (and in particular, imitate all mathematยญ ical proofs). A short digression into model theory will help us to analyze the expresยญ 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


A -- I Introduction -- II Syntax of First-Order Languages -- III Semantics of First-Order Languages -- IV A Sequent Calculus -- V The Completeness Theorem -- VI The Lรถwenheim-Skolem and the Compactness Theorem -- VII The Scope of First-Order Logic -- VIII Syntactic Interpretations and Normal Forms -- B -- IX Extensions of First-Order Logic -- X Limitations of the Formal Method -- XI Free Models and Logic Programming -- XII An Algebraic Characterization of Elementary Equivalence -- XIII Lindstrรถmโ{128}{153}s Theorems -- References -- Symbol Index

Mathematics Mathematical logic Mathematics -- Study and teaching Mathematics Mathematical Logic and Foundations Mathematics Education


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