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Reverse mathematics : proofs from the inside out /
"This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are nee...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton :
Princeton University Press,
[2018]
|
| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 003 | OCoLC | ||
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| 100 | 1 | |a Stillwell, John, |e author. | |
| 245 | 1 | 0 | |a Reverse mathematics : |b proofs from the inside out / |c John Stillwell. |
| 264 | 1 | |a Princeton : |b Princeton University Press, |c [2018] | |
| 264 | 4 | |c ©2018 | |
| 300 | |a 1 online resource (xiii, 182 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | |6 880-01 |a Historical introduction -- Classical arithmetization -- Classical analysis -- Computability -- Arithmetization of computation -- Arithmetical comprehension -- Recursive comprehension -- A bigger picture. | |
| 520 | |a "This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics."-- |c Provided by publisher | ||
| 588 | 0 | |a Online resource; title from electronic title page (EBSCOHost, viewed March 14, 2018). | |
| 590 | |a JSTOR |b Books at JSTOR Evidence Based Acquisitions | ||
| 590 | |a JSTOR |b Books at JSTOR All Purchased | ||
| 590 | |a JSTOR |b Books at JSTOR Demand Driven Acquisitions (DDA) | ||
| 650 | 0 | |a Reverse mathematics. | |
| 650 | 6 | |a Mathématiques à rebours. | |
| 650 | 7 | |a MATHEMATICS |x General. |2 bisacsh | |
| 650 | 7 | |a Reverse mathematics |2 fast | |
| 776 | 0 | 8 | |i Print version: |a Stillwell, John. |t Reverse mathematics. |d Princeton, New Jersey : Princeton University Press, [2018] |z 9780691177175 |w (DLC) 2017025264 |w (OCoLC)983825003 |
| 856 | 4 | 0 | |u https://jstor.uam.elogim.com/stable/10.2307/j.ctvc772m5 |z Texto completo |
| 880 | 0 | 0 | |6 505-01/(S |g Machine generated contents note: |g 1. |t Historical Introduction -- |g 1.1. |t Euclid and the Parallel Axiom -- |g 1.2. |t Spherical and Non-Euclidean Geometry -- |g 1.3. |t Vector Geometry -- |g 1.4. |t Hilbert's Axioms -- |g 1.5. |t Well-ordering and the Axiom of Choice -- |g 1.6. |t Logic and Computability -- |g 2. |t Classical Arithmetization -- |g 2.1. |t From Natural to Rational Numbers -- |g 2.2. |t From Rationals to Reals -- |g 2.3. |t Completeness Properties of R -- |g 2.4. |t Functions and Sets -- |g 2.5. |t Continuous Functions -- |g 2.6. |t Peano Axioms -- |g 2.7. |t Language of PA -- |g 2.8. |t Arithmetically Definable Sets -- |g 2.9. |t Limits of Arithmetization -- |g 3. |t Classical Analysis -- |g 3.1. |t Limits -- |g 3.2. |t Algebraic Properties of Limits -- |g 3.3. |t Continuity and Intermediate Values -- |g 3.4. |t Bolzano-Weierstrass Theorem -- |g 3.5. |t Heine-Borel Theorem -- |g 3.6. |t Extreme Value Theorem -- |g 3.7. |t Uniform Continuity -- |g 3.8. |t Cantor Set -- |g 3.9. |t Trees in Analysis -- |g 4. |t Computability -- |g 4.1. |t Computability and Church's Thesis -- |g 4.2. |t Halting Problem -- |g 4.3. |t Computably Enumerable Sets -- |g 4.4. |t Computable Sequences in Analysis -- |g 4.5. |t Computable Tree with No Computable Path -- |g 4.6. |t Computability and Incompleteness -- |g 4.7. |t Computability and Analysis -- |g 5. |t Arithmetization of Computation -- |g 5.1. |t Formal Systems -- |g 5.2. |t Smullyan's Elementary Formal Systems -- |g 5.3. |t Notations for Positive Integers -- |g 5.4. |t Turing's Analysis of Computation -- |g 5.5. |t Operations on EFS-Generated Sets -- |g 5.6. |t Generating Σ01 Sets -- |g 5.7. |t EFS for Σ01 Relations -- |g 5.8. |t Arithmetizing Elementary Formal Systems -- |g 5.9. |t Arithmetizing Computable Enumeration -- |g 5.10. |t Arithmetizing Computable Analysis -- |g 6. |t Arithmetical Comprehension -- |g 6.1. |t Axiom System ACA0 -- |g 6.2. |t Σ01 and Arithmetical Comprehension -- |g 6.3. |t Completeness Properties in ACA0 -- |g 6.4. |t Arithmetization of Trees -- |g 6.5. |t Konig Infinity Lemma -- |g 6.6. |t Ramsey Theory -- |g 6.7. |t Some Results from Logic -- |g 6.8. |t Peano Arithmetic in ACA0 -- |g 7. |t Recursive Comprehension -- |g 7.1. |t Axiom System RCA0 -- |g 7.2. |t Real Numbers and Continuous Functions -- |g 7.3. |t Intermediate Value Theorem -- |g 7.4. |t Cantor Set Revisited -- |g 7.5. |t From Heine-Borel to Weak Konig Lemma -- |g 7.6. |t From Weak Konig Lemma to Heine-Borel -- |g 7.7. |t Uniform Continuity -- |g 7.8. |t From Weak Konig to Extreme Value -- |g 7.9. |t Theorems of WKL0 -- |g 7.10. |t WKL0, ACA0, and Beyond -- |g 8. |t Bigger Picture -- |g 8.1. |t Constructive Mathematics -- |g 8.2. |t Predicate Logic -- |g 8.3. |t Varieties of Incompleteness -- |g 8.4. |t Computability -- |g 8.5. |t Set Theory -- |g 8.6. |t Concepts of "Depth." |
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