Reverse Mathematics (eBook)

Proofs from the Inside Out

John Stillwell (Autor)

eBook Download: PDF

2018
200 Seiten
Princeton University Press (Verlag)
978-1-4008-8903-7 (ISBN)

Infos

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 "e;right axioms"e; 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 "e;right axiom"e; 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.

Autor

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: From Euclid to Gödel (Princeton).

Produkt-Details

• Erscheint lt. Verlag: 1.1.2018
• Zusatzinfo: 5 halftones. 30 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geschichte der Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften
• Schlagworte: Actual infinity • Affine space • algorithm • Arbitrarily large • associative property • Axiom • Axiom of choice • Big O notation • binary tree • Bolzano–Weierstrass theorem • Bounded quantifier • Cantor's Diagonal Argument • Cantor set • Cantor's Theorem • Choice function • Church–Turing thesis • Classical Logic • coefficient • commutative property • Completeness of the real numbers • Computability • computable function • Computable number • Computation • Constructive analysis • Constructivism (mathematics) • continuous function • Continuous function (set theory) • counterexample • Decision problem • Dedekind cut • definable set • Disjoint sets • Elementary proof • empty set • Entscheidungsproblem • Equation • Extreme value theorem • formal proof • Function (mathematics) • Geometry • Gödel's Incompleteness Theorems • Halting Problem • Heine–Borel theorem • Infimum and supremum • intermediate value theorem • Interval (mathematics) • Invariance of domain • Jordan Curve Theorem • Kruskal's tree theorem • Least-upper-bound property • Logical connective • Mathematical Induction • Mathematics • Maxima and minima • monotone convergence theorem • Monotonic Function • Natural number • Non-Euclidean geometry • number line • Number Theory • Open problem • Pasch's axiom • Peano axioms • pigeonhole principle • polynomial • Predicate logic • Prime number • prime number theorem • Pythagorean Theorem • Quantifier (logic) • Ramsey's theorem • Ramsey theory • Rational number • real number • Recursively enumerable set • reverse mathematics • Riemann Mapping Theorem • scientific notation • Sequence • Set (mathematics) • set theory • Significant figures • Sign (mathematics) • Special case • Sperner's Lemma • Subset • Successor function • Summation • Theorem • turing degree • Turing jump • Turing Machine • uniform continuity • Upper and lower bounds • Variable (mathematics) • Well-order • Well-ordering theorem • Without loss of generality • Zermelo–Fraenkel set theory
• ISBN-10: 1-4008-8903-0 / 1400889030
• ISBN-13: 978-1-4008-8903-7 / 9781400889037