Gödel's Incompleteness Theorems are among the most significant results in the foundation of mathematics. These results have a positive consequence: any system of axioms for mathematics that we recognize as correct can be properly extended by adding as a new axiom a formal statement expressing that the original system is consistent. This suggests that our mathematical knowledge is inexhaustible, an essentially philosophical topic to which this book is devoted. Basic material in predicate logic, set theory and recursion theory is presented, leading to a proof of incompleteness theorems. The inexhaustibility of mathematical knowledge is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman. All concepts and results necessary to understand the arguments are introduced as needed, making the presentation self-contained and thorough.
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"In this book the author discusses Gödel's famous incompleteness theorems. Special emphasis is put on the consequences of the inexhaustibility of our mathematical knowledge in any one formal axiomatic theory ... The book can be considered as a more technical companion to the author's more philosophical book [Gödel's Theorem, A K Peters, Wellesley, MA, 2005]." -Mathematiacl Reviews, November 2007
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In 1931, Kurt Godel presented his famous incompleteness theorem, which has since had a great and continuing impact on logic and the philosophy of mathematics, and has also like no other result in formal logic caught the interest and imagination of the general public. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
actual provability, impredicative comprehension principle, purely arithmetical proof, local reflection principle, uniform reflection principle, true arithmetical sentences, truth extensions, ordinary mathematical reasoning, formal provability, computable partial function, deducibility relation, autonomous sequence, qnxn where, iterated extensions, arithmetical statement, recursion theorem, arithmetical language, ordinal notations, arithmetical theorems, arithmetical proofs, arithmetical knowledge, arithmetical truth, primitive recursive, set quantifiers, being provable New!
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Front Cover | Table of Contents | First Pages | Index | Surprise Me!
In 1931, Kurt Godel presented his famous incompleteness theorem, which has since had a great and continuing impact on logic and the philosophy of mathematics, and has also like no other result in formal logic caught the interest and imagination of the general public. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
actual provability, impredicative comprehension principle, purely arithmetical proof, local reflection principle, uniform reflection principle, true arithmetical sentences, truth extensions, ordinary mathematical reasoning, formal provability, computable partial function, deducibility relation, autonomous sequence, qnxn where, iterated extensions, arithmetical statement, recursion theorem, arithmetical language, ordinal notations, arithmetical theorems, arithmetical proofs, arithmetical knowledge, arithmetical truth, primitive recursive, set quantifiers, being provable New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
Citations (learn more)
This book cites 5
books:
- Model Theory, Third Edition (Studies in Logic and the Foundations of Mathematics) by C.C. Chang in Back Matter
- Mathematical Logic by Joseph R. Shoenfield in Back Matter
- Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman, Lecture Notes in Logic 15 by Wilfried Sieg in Back Matter
- The Czech Republic: The Velvet Revolution (Arbitrary Borders) by Robert Charles Cottrell in Back Matter
- Mathematical logic by W. V Quine in Back Matter
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