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.
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