Two Applications of Logic to Mathematics (eBook)

Gaisi Takeuti (Autor)

eBook Download: PDF

2015 | 1. Auflage
148 Seiten
Princeton University Press (Verlag)
978-1-4008-7134-6 (ISBN)

Infos

Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing that any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem. Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs.Originally published in 1978.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Produkt-Details

• Erscheint lt. Verlag: 8.3.2015
• Reihe/Serie: Princeton Legacy Library
• Reihe/Serie: Princeton Legacy Library
• Reihe/Serie: Publications of the Mathematical Society of Japan
• Reihe/Serie: Publications of the Mathematical Society of Japan
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Schlagworte: analytic continuation • analytic function • arithmetic • arithmetic progression • automorphism • Axiom • Baire category theorem • Baire function • Banach space • bisection method • Boolean algebra • Boolean algebra (structure) • Borel set • bounded operator • Calculation • Cauchy–Riemann equations • Cauchy Sequence • Cauchy's Integral Formula • Cauchy's theorem (geometry) • Cauchy's theorem (group theory) • Characteristic function (probability theory) • Classical Logic • commutative property • Complete Boolean algebra • Complex Analysis • complex number • Connected space • conservative extension • continuous function • Continuous function (set theory) • counterexample • Dedekind cut • differential equation • Dimensional Analysis • Equation • Euclidean space • existential quantification • Fubini's Theorem • Function composition • Function (mathematics) • hilbert space • Implicit function • inference • Interval (mathematics) • Laplace's equation • Lebesgue measure • Limit of a sequence • Limit point • Limit superior and limit inferior • Linear space (geometry) • Line (geometry) • Mathematical Analysis • Mathematical Induction • Mathematical Logic • mathematical practice • Mathematics • Maximum principle • Measurable function • Measure (mathematics) • multiplication operator • Natural number • Number Theory • Order theory • ordinal number • partition of unity • Peano axioms • Predicate (mathematical logic) • prime number theorem • Primitive Recursive Arithmetic • Primitive Recursive Function • product measure • Projection (linear algebra) • Proof theory • Quantifier (logic) • Quantum Logic • Rational number • Real analysis • real number • Residue Theorem • Riemann Mapping Theorem • Riemann surface • scientific notation • Self-adjoint • Self-adjoint operator • sequent • set theory • Simply connected space • Special case • step function • Taylor's theorem • Theorem • Three-dimensional space (mathematics) • topological group • Topological space • Topology • Transfinite induction • Type Theory • Variable (mathematics) • Weierstrass Theorem • Zermelo–Fraenkel set theory
• ISBN-10: 1-4008-7134-4 / 1400871344
• ISBN-13: 978-1-4008-7134-6 / 9781400871346