{"title":"Heinz-Dieter Ebbinghaus","description":null,"products":[{"product_id":"mathematical-logic-book-heinz-dieter-ebbinghaus-9783030738389","title":"Mathematical Logic","description":"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.","brand":"WoB","offers":[{"title":"GB \/ NEW \/ GARDNERS","offer_id":49739309941009,"sku":"NGR9783030738389","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":50281266348305,"sku":"CIN3030738388VG","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ GOOD \/ SBYB","offer_id":53068875923729,"sku":"CIN3030738388G","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3030738388.jpg?v=1751439380"},{"product_id":"numbers-book-heinz-dieter-ebbinghaus-9780387974972","title":"Numbers","description":"A book about numbers sounds rather dull. This one is not. Instead it is a lively story about one thread of mathematics-the concept of number- told by eight authors and organized into a historical narrative that leads the reader from ancient Egypt to the late twentieth century. It is a story that begins with some of the simplest ideas of mathematics and ends with some of the most complex. It is a story that mathematicians, both amateur and professional, ought to know. Why write about numbers? Mathematicians have always found it diffi- cult to develop broad perspective about their subject. While we each view our specialty as having roots in the past, and sometimes having connec- tions to other specialties in the present, we seldom see the panorama of mathematical development over thousands of years. Numbers attempts to give that broad perspective, from hieroglyphs to K-theory, from Dedekind cuts to nonstandard analysis.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50351599223057,"sku":"CIN0387974970G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":51894150496529,"sku":"GOR009644945","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52135113982225,"sku":"NLS9780387974972","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/B00C9AXCY0.jpg?v=1750878703"},{"product_id":"mathematical-logic-book-heinz-dieter-ebbinghaus-9783030738419","title":"Mathematical Logic","description":"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. 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Begriffe wie \"reelle oder komplexe Zahlen, Pi\" sind zwar jedem geläufig, aber nur wenige wissen, was sich wirklich dahinter verbirgt. Die Autoren dieses Bandes geben jedem, der mehr wissen will als nur die Hülle der Begriffe, eine meisterhafte Einführung in die Magie der Mathematik und schlagen einzigartige Brücken für Studenten.  Die Rezensenten der ersten beiden Auflagen überschlugen sich.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51118267072785,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":51118268875025,"sku":"NIN9783540556541","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52141869695249,"sku":"NLS9783540556541","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/B002DONB20.jpg?v=1750807215"},{"product_id":"finite-model-theory-book-heinz-dieter-ebbinghaus-9783540287872","title":"Finite Model Theory","description":"Finite model theory, the model theory of finite structures, has roots in clas- sical model theory; however, its systematic development was strongly influ- enced by research and questions of complexity theory and of database theory. Model theory or the theory of models, as it was first named by Tarski in 1954, may be considered as the part of the semantics of formalized languages that is concerned with the interplay between the syntactic structure of an axiom system on the one hand and (algebraic, settheoretic, . . . ) properties of its models on the other hand. As it turned out, first-order language (we mostly speak of first-order logic) became the most prominent language in this respect, the reason being that it obeys some fundamental principles such as the compactness theorem and the completeness theorem. These principles are valuable modeltheoretic tools and, at the same time, reflect the expressive weakness of first-order logic. This weakness is the breeding ground for the freedom which modeltheoretic methods rest upon. By compactness, any first-order axiom system either has only finite models of limited cardinality or has infinite models. The first case is trivial because finitely many finite structures can explicitly be described by a first-order sentence. As model theory usually considers all models of an axiom system, modeltheorists were thus led to the second case, that is, to infinite structures. In fact, classical model theory of first-order logic and its generalizations to stronger languages live in the realm of the infinite.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51457681293585,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":51457681588497,"sku":"GOR014287606","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52434641944849,"sku":"NLS9783540287872","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/B01DWUB8NW.jpg?v=1753786519"},{"product_id":"recursion-theory-week-book-heinz-dieter-ebbinghaus-9783540156734","title":"Recursion Theory Week","description":null,"brand":"WoB","offers":[{"title":"GB \/ NEW \/ INGRAM","offer_id":52135957528849,"sku":"NLS9783540156734","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9783540156734.jpg?v=1757551848"},{"product_id":"ernst-zermelo-book-heinz-dieter-ebbinghaus-9783642080500","title":"Ernst Zermelo","description":"This biography attempts to shed light on all facets of Zermelo's life and achievements. 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