##
**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

### 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 |