Two Applications of Logic to Mathematics

Gaisi Takeuti (Autor)

Buch | Softcover

148 Seiten

2015
Princeton University Press (Verlag)
978-0-691-61022-1 (ISBN)

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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.
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*Frontmatter, pg. i*Preface, pg. v*Contents, pg. vii*Introduction, pg. 1*Introduction, pg. 5*Chapter 1. Boolean Valued Analysis Using Projection Algebras, pg. 6*Chapter 2. Boolean Valued Analysis Using Measure Algebras, pg. 51*References, pg. 71*Introduction, pg. 73*Chapter 1. Real Analysis, pg. 77*Chapter 2. Complex Analysis, pg. 114*References, pg. 136*Index, pg. 138

Erscheint lt. Verlag	8.3.2015
Reihe/Serie	Publications of the Mathematical Society of Japan
Verlagsort	New Jersey
Sprache	englisch
Maße	152 x 235 mm
Gewicht	198 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
ISBN-10	0-691-61022-3 / 0691610223
ISBN-13	978-0-691-61022-1 / 9780691610221
Zustand	Neuware