Robinson, Abraham Non-standard analysis. (English) Zbl 0151.00803 Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Company. xi, 293 p. (1966). The author delights in showing that the old non-rigorous approach to analysis by means of infinitesimals can, in many respects, be justified and expanded by use of non-standard models for analysis. The latter are models which satisfy the same elementary properties (i.e. those expressible in the first-order predicate calculus) as the real number system but are not isomorphic to that system. Topics covered include: tools from logic, differential and integral calculus, general topology, functions of real and complex variables, linear spaces, topological groups and Lie groups, variational problems, hydrodynamics, and the history of the calculus. A striking application is the solution by A. R. Bernstein and the author [Pac. J. Math. 16, 421–431 (1966; Zbl 0141.12903)] of an open invariant subspace problem of P. R. Halmos and K. T. Smith. Reviewer: Elliott Mendelson Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 16 ReviewsCited in 304 Documents MathOverflow Questions: Is there a source linking Robinson’s work in wing theory with his theory of infinitesimals? MSC: 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03Hxx Nonstandard models 26E35 Nonstandard analysis 26-02 Research exposition (monographs, survey articles) pertaining to real functions 26-03 History of real functions 01A65 Development of contemporary mathematics 28E05 Nonstandard measure theory 30G06 Non-Archimedean function theory 46S20 Nonstandard functional analysis 47S20 Nonstandard operator theory 54J05 Nonstandard topology Keywords:mathematical logic; nonstandard analysis; applications; topology, functions of a real variable; functions of a complex variable; normed linear spaces; boundary layer flow of viscous fluids; rederivations of Saint-Venant’s hypothesis; distribution of stresses in an elastic body Citations:Zbl 0141.12903 × Cite Format Result Cite Review PDF