Two Applications of Logic to Mathematics

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The book explores the application of mathematical logic through two distinct approaches: set theory and proof theory. In the first part, Gaisi Takeuti utilizes Scott-Solovay's Boolean-valued models to enhance analysis with complete Boolean algebras of projections. The second part focuses on classical and complex analysis within Peano's arithmetic, demonstrating that any theorem from analytic number theory can also be derived in Peano's framework, employing Gentzen's cut elimination theorem to strengthen the connections between these mathematical disciplines.