本世纪最好的NSA！ 

ONSTANDARD ANALYSIS

By DR. J. PONSTEIN

With love, love, love to those ve women, who caressed me.

J. Ponstein former professor at the University of Groningen

A Naive Way to the Innitesimals (an unorthodox treatment of Nonstandard Analysis)

ISBN: 90-367-1672-1

Contents

Prologue 11

Preface 15

1 Generalities 17

1.1 Innitesimals and other nonstandard numbers: getting acquainted 17

1.2 Other ∗-transforms; generating new numbers . . . . . . . . . . . . 19 1.3 Bound and free variables; prenex normal form . . . . . . . . . . . 20

1.4 The purpose of nonstandard analysis . . . . . . . . . . . . . . . . 23

1.5 More about the ∗-transform; transfer . . . . . . . . . . . . . . . . 25

1.6 Standard, internal, and external constants . . . . . . . . . . . . . 28

1.7 Innitesimals in Greek geometry? . . . . . . . . . . . . . . . . . . 29

1.8 Innitesimals in the 17th to the 19th century . . . . . . . . . . . . 31

1.9 Innitesimals in the 20th century . . . . . . . . . . . . . . . . . . 34

1.10 Introducing innitesimals by plausible reasoning; lters . . . . . . 37

1.11 Basic assumptions of formalism . . . . . . . . . . . . . . . . . . . 41

1.12 Basic assumptions of constructivism . . . . . . . . . . . . . . . . . 43

1.13 Selecting basic assumptions naively . . . . . . . . . . . . . . . . . 45

1.14 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.15 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.16 About the nature of free ultralters . . . . . . . . . . . . . . . . . 53

2 Basic theory 55

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