Two Applications of Logic to Mathematics
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
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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Mathematical practice, Topology, Riemann mapping theorem, Cauchy's integral formula, Self-adjoint, Characteristic function (probability theory), Multiplication operator, Function (mathematics), Theorem, Complete Boolean algebra, Quantum logic, Analytic continuation, Sequent, Classical logic, Boolean algebra, Primitive recursive function, Bounded operator, Primitive recursive arithmetic, Measurable function, Fubini's theorem, Linear space (geometry), Cauchy's theorem (geometry), Limit superior and limit inferior, Existential quantification, Proof theory, Arithmetic, Transfinite induction, Arithmetic progression, Boolean algebra (structure), Mathematical logic, Order theory, Self-adjoint operator, Continuous function (set theory), Baire category theorem, Weierstrass theorem, Commutative property, Real number, Analytic function, Complex number, Real analysis, Axiom, Riemann surface, Interval (mathematics), Topological space, Partition of unity, Mathematics, Predicate (mathematical logic), Zermelo–Fraenkel set theory, Cauchy's theorem (group theory), Complex analysis, Set theory, Number theory, Measure (mathematics), Continuous function, Mathematical analysis, Mathematical induction, Hilbert space, Natural number, Lebesgue measure, Automorphism, Cauchy–Riemann equations, Line (geometry), Peano axioms, Projection (linear algebra), Rational number, Taylor's theorem, Type theory, Baire function, Conservative extension, Quantifier (logic)