04073nam 22006135 450 991048073210332120210915185544.01-4757-2355-510.1007/978-1-4757-2355-7(CKB)2660000000024224(SSID)ssj0001297533(PQKBManifestationID)11725945(PQKBTitleCode)TC0001297533(PQKBWorkID)11229082(PQKB)11271152(DE-He213)978-1-4757-2355-7(MiAaPQ)EBC3084680(PPN)238077004(EXLCZ)99266000000002422420130109d1994 u| 0engurnn#008mamaatxtccrMathematical Logic[electronic resource] /by H.-D. Ebbinghaus, J. Flum, Wolfgang Thomas2nd ed. 1994.New York, NY :Springer New York :Imprint: Springer,1994.1 online resource (X, 291 p.)Undergraduate Texts in Mathematics,0172-6056Bibliographic Level Mode of Issuance: Monograph0-387-94258-0 1-4757-2357-1 Includes bibliographical references and indexes.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 Löwenheim-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 Lindström’s Theorems -- References -- Symbol Index.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.Undergraduate Texts in Mathematics,0172-6056Mathematical logicMathematics—Study and teaching Mathematical Logic and Foundationshttps://scigraph.springernature.com/ontologies/product-market-codes/M24005Mathematics Educationhttps://scigraph.springernature.com/ontologies/product-market-codes/O25000Mathematical logic.Mathematics—Study and teaching .Mathematical Logic and Foundations.Mathematics Education.511.303-01mscEbbinghaus H.-Dauthttp://id.loc.gov/vocabulary/relators/aut1045262Flum Jauthttp://id.loc.gov/vocabulary/relators/autThomas Wolfgangauthttp://id.loc.gov/vocabulary/relators/autBOOK9910480732103321Mathematical Logic2471401UNINA