LEADER 04088nam 22006135 450 001 9910792487603321 005 20230825134723.0 010 $a1-4757-2355-5 024 7 $a10.1007/978-1-4757-2355-7 035 $a(CKB)2660000000024224 035 $a(SSID)ssj0001297533 035 $a(PQKBManifestationID)11725945 035 $a(PQKBTitleCode)TC0001297533 035 $a(PQKBWorkID)11229082 035 $a(PQKB)11271152 035 $a(DE-He213)978-1-4757-2355-7 035 $a(MiAaPQ)EBC3084680 035 $a(PPN)238077004 035 $a(EXLCZ)992660000000024224 100 $a20130109d1994 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aMathematical Logic$b[electronic resource] /$fby H.-D. Ebbinghaus, J. Flum, Wolfgang Thomas 205 $a2nd ed. 1994. 210 1$aNew York, NY :$cSpringer New York :$cImprint: Springer,$d1994. 215 $a1 online resource (X, 291 p.) 225 1 $aUndergraduate Texts in Mathematics,$x0172-6056 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a0-387-94258-0 311 $a1-4757-2357-1 320 $aIncludes bibliographical references and indexes. 327 $aA -- 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. 330 $aWhat is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathe­ matical proofs? Only in this century has there been success in obtaining substantial and satisfactory answers. The present book contains a systematic discussion of these results. The investigations are centered around first-order logic. Our first goal is Godel's completeness theorem, which shows that the con­ sequence relation coincides with formal provability: By means of a calcu­ lus consisting of simple formal inference rules, one can obtain all conse­ quences of a given axiom system (and in particular, imitate all mathemat­ ical proofs). A short digression into model theory will help us to analyze the expres­ sive power of the first-order language, and it will turn out that there are certain deficiencies. For example, the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis. On the other hand, this difficulty can be overcome--even in the framework of first-order logic-by developing mathematics in set-theoretic terms. We explain the prerequisites from set theory necessary for this purpose and then treat the subtle relation between logic and set theory in a thorough manner. 410 0$aUndergraduate Texts in Mathematics,$x0172-6056 606 $aMathematical logic 606 $aMathematics?Study and teaching  606 $aMathematical Logic and Foundations$3https://scigraph.springernature.com/ontologies/product-market-codes/M24005 606 $aMathematics Education$3https://scigraph.springernature.com/ontologies/product-market-codes/O25000 615 0$aMathematical logic. 615 0$aMathematics?Study and teaching . 615 14$aMathematical Logic and Foundations. 615 24$aMathematics Education. 676 $a511.3 686 $a03-01$2msc 700 $aEbbinghaus$b Heinz-Dieter$f1939-$4aut$4http://id.loc.gov/vocabulary/relators/aut$01068399 702 $aFlum$b J$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aThomas$b Wolfgang$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910792487603321 996 $aMathematical logic$92553144 997 $aUNINA