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Reverse Mathematics: Proofs from the Inside Out [Pehme köide]

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  • Formaat: Paperback / softback, 200 pages, kõrgus x laius: 235x156 mm, 35 b/w illus.
  • Ilmumisaeg: 24-Sep-2019
  • Kirjastus: Princeton University Press
  • ISBN-10: 0691196419
  • ISBN-13: 9780691196411
Teised raamatud teemal:
Reverse Mathematics: Proofs from the Inside OutSuurem pilt
  • Formaat: Paperback / softback, 200 pages, kõrgus x laius: 235x156 mm, 35 b/w illus.
  • Ilmumisaeg: 24-Sep-2019
  • Kirjastus: Princeton University Press
  • ISBN-10: 0691196419
  • ISBN-13: 9780691196411
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780691196411.html
Märksõnad:

The first book surveying the history and ideas behind reverse mathematics

Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics, John Stillwell offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. 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.

Arvustused

"The field has been due for a general treatment accessible to undergraduates and to mathematicians in other areas. . . . With Reverse Mathematics, John Stillwell provides exactly that kind of introduction."Carl Mummert, Notices of the American Mathematical Society "Stillwell carefully situates the field in the broader context of the history of mathematics and its foundations, and does a fine job of making the whole endeavor accessible to a general mathematical audience."Jeremy Avigad, Carnegie Mellon University "Filling an important niche, this book gives readers a good picture of the basics of reverse mathematics while suggesting several directions for further reading and study."Denis Hirschfeldt, University of Chicago "Stillwell's book is self-contained and includes much background material in analysis, mathematical logic, combinatorics, and computability. I heartily commend this very readable and accessible book."Stephen Simpson, Vanderbilt University

Preface xi
1 Historical Introduction
1(25)
1.1 Euclid and the Parallel Axiom
2(3)
1.2 Spherical and Non-Euclidean Geometry
5(5)
1.3 Vector Geometry
10(4)
1.4 Hilbert's Axioms
14(5)
1.5 Well-ordering and the Axiom of Choice
19(4)
1.6 Logic and Computability
23(3)
2 Classical Arittimetization
26(25)
2.1 From Natural to Rational Numbers
27(2)
2.2 From Rationals to Reals
29(3)
2.3 Completeness Properties of R
32(3)
2.4 Functions and Sets
35(2)
2.5 Continuous Functions
37(2)
2.6 The Peano Axioms
39(4)
2.7 The Language of PA
43(2)
2.8 Arithmetically Definable Sets
45(3)
2.9 Limits of Arithmetization
48(3)
3 Classical Analysis
51(19)
3.1 Limits
51(2)
3.2 Algebraic Properties of Limits
53(2)
3.3 Continuity and Intermediate Values
55(2)
3.4 The Bolzano-Weierstrass Theorem
57(2)
3.5 The Heine-Borel Theorem
59(1)
3.6 The Extreme Value Theorem
60(1)
3.7 Uniform Continuity
61(3)
3.8 The Cantor Set
64(2)
3.9 Trees in Analysis
66(4)
4 Computability
70(15)
4.1 Computability and Church's Thesis
71(2)
4.2 The Halting Problem
73(1)
4.3 Computably Enumerable Sets
74(3)
4.4 Computable Sequences in Analysis
77(1)
4.5 Computable Tree with No Computable Path
78(2)
4.6 Computability and Incompleteness
80(1)
4.7 Computability and Analysis
81(4)
5 Arithmetization of Computation
85(24)
5.1 Formal Systems
86(1)
5.2 Smullyan's Elementary Formal Systems
87(2)
5.3 Notations for Positive Integers
89(2)
5.4 Turing's Analysis of Computation
91(2)
5.5 Operations on EFS-Generated Sets
93(3)
5.6 Generating Zj Sets
96(2)
5.7 EFS for I? Relations
98(2)
5.8 Arithmetizing Elementary Formal Systems
100(3)
5.9 Arithmetizing Computable Enumeration
103(3)
5.10 Arithmetizing Computable Analysis
106(3)
6 Arithmetical Comprehension
109(21)
6.1 The Axiom System ACA0
110(1)
6.2 Z and Arithmetical Comprehension
111(2)
6.3 Completeness Properties in ACA0
113(3)
6.4 Arithmetization of Trees
116(2)
6.5 The Konig Infinity Lemma
118(3)
6.6 Ramsey Theory
121(3)
6.7 Some Results from Logic
124(3)
6.8 Peano Arithmetic in ACA0
127(3)
7 Recursive Comprehension
130(24)
7.1 The Axiom System RCA0
131(1)
7.2 Real Numbers and Continuous Functions
132(2)
7.3 The Intermediate Value Theorem
134(2)
7.4 The Cantor Set Revisited
136(1)
7.5 From Heine-Borel to Weak Konig Lemma
137(3)
7.6 From Weak Konig Lemma to Heine-Borel
140(1)
7.7 Uniform Continuity
141(2)
7.8 From Weak Kdnig to Extreme Value
143(3)
7.9 Theorems of WKL0
146(3)
7.10 WKL0, ACA0, and Beyond
149(5)
8 A Bigger Picture
154(14)
8.1 Constructive Mathematics
155(1)
8.2 Predicate Logic
156(3)
8.3 Varieties of Incompleteness
159(3)
8.4 Computability
162(2)
8.5 Set Theory
164(2)
8.6 Concepts of "Depth"
166(2)
Bibliography 168(5)
Index 173
John Stillwell is professor of mathematics at the University of San Francisco and an affiliate of the School of Mathematical Sciences at Monash University, Australia. His many books include Mathematics and Its History and Elements of Mathematics (Princeton).