{"product_id":"9780387942582","title":"Undergraduate Texts in Mathematics","description":"\u003ch1\u003eUndergraduate Texts in Mathematics\u003c\/h1\u003e \u003ch2\u003eEbbinghaus, H.-D.; Flum, J.; Thomas, Wolfgang\u003c\/h2\u003e \u003cp\u003eWhat 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.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 1994-06-10\u003c\/p\u003e \u003cp\u003eFormat: Hardcover\u003c\/p\u003e \u003cp\u003eISBN-13: 9780387942582\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-1-4757-2355-7\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 291\u003c\/p\u003e ","brand":"Springer New York","offers":[{"title":"Default Title","offer_id":44398570504332,"sku":"9780387942582","price":71.95,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9780387942582.jpg?v=1772815343","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9780387942582","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}