An introduction to Gödel's Theorems:
In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also...
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2007
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| Series: | Cambridge introductions to philosophy
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| Subjects: | |
| Online Access: | BSB01 FHN01 UBG01 Volltext |
| Summary: | In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic |
| Item Description: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Physical Description: | 1 online resource (xiv, 361 pages) |
| ISBN: | 9780511800962 |
| DOI: | 10.1017/CBO9780511800962 |
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Record in the Search Index
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| any_adam_object | |
| author | Smith, Peter 1944- |
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| contents | What Godel's theorems say -- Decidability and enumerability -- Axiomatized formal theories -- Capturing numerical properties -- The truths of arithmetic -- Sufficiently strong arithmetics -- Interlude: taking stock -- Two formalized arithmetics -- What q can prove -- First-order peano arithmetic -- Primitive recursive functions -- Capturing p r functions -- Q is p.r. adequate -- Interlude: a very little about Principia -- The arithmetization of syntax -- PA is incomplete -- Godel's first theorem -- Interlude: about the first theorem -- Strengthening the first theorem -- The diagonalization lemma -- Using the diagonalization lemma -- Second-order arithmetics -- Interlude: incompleteness and Isaacson's conjecture -- Godel's second theorem for PA -- The derivability conditions -- Deriving the derivability conditions -- Reflections -- Interlude: about the second theorem -- Recursive functions -- Undecidability and incompleteness -- Turing machines -- Turing machines and recursiveness -- Halting problems -- The church-turing thesis -- Proving the thesis |
| ctrlnum | (ZDB-20-CBO)CR9780511800962 (OCoLC)971456813 (DE-599)BVBBV043919652 |
| 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 |
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| discipline | Mathematik Philosophie |
| doi_str_mv | 10.1017/CBO9780511800962 |
| format | Electronic eBook |
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| isbn | 9780511800962 |
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| spelling | Smith, Peter 1944- Verfasser (DE-588)173658121 aut An introduction to Gödel's Theorems Peter Smith Cambridge Cambridge University Press 2007 1 online resource (xiv, 361 pages) txt rdacontent c rdamedia cr rdacarrier Cambridge introductions to philosophy Title from publisher's bibliographic system (viewed on 05 Oct 2015) What Godel's theorems say -- Decidability and enumerability -- Axiomatized formal theories -- Capturing numerical properties -- The truths of arithmetic -- Sufficiently strong arithmetics -- Interlude: taking stock -- Two formalized arithmetics -- What q can prove -- First-order peano arithmetic -- Primitive recursive functions -- Capturing p r functions -- Q is p.r. adequate -- Interlude: a very little about Principia -- The arithmetization of syntax -- PA is incomplete -- Godel's first theorem -- Interlude: about the first theorem -- Strengthening the first theorem -- The diagonalization lemma -- Using the diagonalization lemma -- Second-order arithmetics -- Interlude: incompleteness and Isaacson's conjecture -- Godel's second theorem for PA -- The derivability conditions -- Deriving the derivability conditions -- Reflections -- Interlude: about the second theorem -- Recursive functions -- Undecidability and incompleteness -- Turing machines -- Turing machines and recursiveness -- Halting problems -- The church-turing thesis -- Proving the thesis In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic Gödel, Kurt Gödel numbers Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 s 2\p DE-604 Erscheint auch als Druckausgabe 978-0-521-67453-9 Erscheint auch als Druckausgabe 978-0-521-85784-0 https://doi.org/10.1017/CBO9780511800962 Verlag URL des Erstveröffentlichers 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 |
| spellingShingle | Smith, Peter 1944- An introduction to Gödel's Theorems What Godel's theorems say -- Decidability and enumerability -- Axiomatized formal theories -- Capturing numerical properties -- The truths of arithmetic -- Sufficiently strong arithmetics -- Interlude: taking stock -- Two formalized arithmetics -- What q can prove -- First-order peano arithmetic -- Primitive recursive functions -- Capturing p r functions -- Q is p.r. adequate -- Interlude: a very little about Principia -- The arithmetization of syntax -- PA is incomplete -- Godel's first theorem -- Interlude: about the first theorem -- Strengthening the first theorem -- The diagonalization lemma -- Using the diagonalization lemma -- Second-order arithmetics -- Interlude: incompleteness and Isaacson's conjecture -- Godel's second theorem for PA -- The derivability conditions -- Deriving the derivability conditions -- Reflections -- Interlude: about the second theorem -- Recursive functions -- Undecidability and incompleteness -- Turing machines -- Turing machines and recursiveness -- Halting problems -- The church-turing thesis -- Proving the thesis Gödel, Kurt Gödel numbers Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd |
| subject_GND | (DE-588)4021417-5 (DE-588)4151278-9 |
| title | An introduction to Gödel's Theorems |
| title_auth | An introduction to Gödel's Theorems |
| title_exact_search | An introduction to Gödel's Theorems |
| title_full | An introduction to Gödel's Theorems Peter Smith |
| title_fullStr | An introduction to Gödel's Theorems Peter Smith |
| title_full_unstemmed | An introduction to Gödel's Theorems Peter Smith |
| title_short | An introduction to Gödel's Theorems |
| title_sort | an introduction to godel s theorems |
| topic | Gödel, Kurt Gödel numbers Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd |
| topic_facet | Gödel, Kurt Gödel numbers Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz Einführung |
| url | https://doi.org/10.1017/CBO9780511800962 |
| work_keys_str_mv | AT smithpeter anintroductiontogodelstheorems |