Mathematical Logic:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1994
|
| Ausgabe: | Second Edition |
| Schriftenreihe: | Undergraduate Texts in Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical 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 consequence relation coincides with formal provability: By means of a calculus consisting of simple formal inference rules, one can obtain all consequences of a given axiom system (and in particular, imitate all mathematical proofs). A short digression into model theory will help us to analyze the expressive 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 |
| Beschreibung: | 1 Online-Ressource (X, 291 p) |
| ISBN: | 9781475723557 9781475723571 |
| ISSN: | 0172-6056 |
| DOI: | 10.1007/978-1-4757-2355-7 |
Internformat
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| 250 | |a Second Edition | ||
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| 500 | |a What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical 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 consequence relation coincides with formal provability: By means of a calculus consisting of simple formal inference rules, one can obtain all consequences of a given axiom system (and in particular, imitate all mathematical proofs). A short digression into model theory will help us to analyze the expressive 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 | ||
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Datensatz im Suchindex
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| author | Ebbinghaus, H.-D |
| author_facet | Ebbinghaus, H.-D |
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| author_sort | Ebbinghaus, H.-D |
| author_variant | h d e hde |
| building | Verbundindex |
| bvnumber | BV042421334 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
| ctrlnum | (OCoLC)1165440225 (DE-599)BVBBV042421334 |
| dewey-full | 511.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3 |
| dewey-search | 511.3 |
| dewey-sort | 3511.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4757-2355-7 |
| edition | Second Edition |
| format | Electronic eBook |
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| language | English |
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| spelling | Ebbinghaus, H.-D. Verfasser aut Mathematical Logic by H.-D. Ebbinghaus, J. Flum, W. Thomas Second Edition New York, NY Springer New York 1994 1 Online-Ressource (X, 291 p) txt rdacontent c rdamedia cr rdacarrier Undergraduate Texts in Mathematics 0172-6056 What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical 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 consequence relation coincides with formal provability: By means of a calculus consisting of simple formal inference rules, one can obtain all consequences of a given axiom system (and in particular, imitate all mathematical proofs). A short digression into model theory will help us to analyze the expressive 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 Mathematics Logic, Symbolic and mathematical Mathematical Logic and Foundations Mathematik Mathematische Logik (DE-588)4037951-6 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content 2\p (DE-588)4123623-3 Lehrbuch gnd-content Mathematische Logik (DE-588)4037951-6 s 3\p DE-604 Flum, J. Sonstige oth Thomas, W. Sonstige oth https://doi.org/10.1007/978-1-4757-2355-7 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ebbinghaus, H.-D Mathematical Logic Mathematics Logic, Symbolic and mathematical Mathematical Logic and Foundations Mathematik Mathematische Logik (DE-588)4037951-6 gnd |
| subject_GND | (DE-588)4037951-6 (DE-588)4151278-9 (DE-588)4123623-3 |
| title | Mathematical Logic |
| title_auth | Mathematical Logic |
| title_exact_search | Mathematical Logic |
| title_full | Mathematical Logic by H.-D. Ebbinghaus, J. Flum, W. Thomas |
| title_fullStr | Mathematical Logic by H.-D. Ebbinghaus, J. Flum, W. Thomas |
| title_full_unstemmed | Mathematical Logic by H.-D. Ebbinghaus, J. Flum, W. Thomas |
| title_short | Mathematical Logic |
| title_sort | mathematical logic |
| topic | Mathematics Logic, Symbolic and mathematical Mathematical Logic and Foundations Mathematik Mathematische Logik (DE-588)4037951-6 gnd |
| topic_facet | Mathematics Logic, Symbolic and mathematical Mathematical Logic and Foundations Mathematik Mathematische Logik Einführung Lehrbuch |
| url | https://doi.org/10.1007/978-1-4757-2355-7 |
| work_keys_str_mv | AT ebbinghaushd mathematicallogic AT flumj mathematicallogic AT thomasw mathematicallogic |