Nonstandard analysis

Contents......Page 7
Prologue......Page 11
Preface......Page 17
1.1 Infinitesimals and other nonstandard numbers:getting acquainted......Page 19
1.2 Other *-transforms; generating new numbers......Page 21
1.3 Bound and free variables; prenex normal form......Page 22
1.4 The purpose of nonstandard analysis......Page 25
1.5 More about the *-transform; transfer......Page 27
1.6 Standard, internal, and external constants......Page 30
1.7 Infinitesimals in Greek geometry?......Page 31
1.8 Infinitesimals in the 17th to the 19th century......Page 33
1.9 Infinitesimals in the 20th century......Page 36
1.10 Introducing infinitesimals by plausible reasonIng;filters......Page 39
1.11 Basic assumptions of formalism......Page 43
1.12 Basic assumptions of constructivism......Page 45
1.13 Selecting basic assumptions naively......Page 47
1.14 Basic definitions......Page 51
1.15 Filters......Page 53
1.16 About the nature of free ultrafilters......Page 55
2.1 Reviewing the introduction of Z, 2.2 Introducing internal constants; definition of equality......Page 59
2.3 Identification of internal constants......Page 60
2.4 Standard constants; basic results for internalconstants......Page 65
2.5 External constants......Page 69
2.6 The *-transform of operations and expressions......Page 71
2.7 The *-transform of relations and statements; Los'theorem; the internal definition principle......Page 73
2.8 Transfer; the standard definition principle......Page 80
2.9 The *-transform of attributes......Page 83
2.10 *N, *~, "(Q, *1R: main definitions and properties......Page 85
2.11 Overflow and underflow......Page 87
2.12 *N and *Z: more properties......Page 89
2.13 *2.14 An alternative to introducing *Z, "(Q and *1R......Page 94
2.15 Getting away with generating sequences and H(Si); summary......Page 96
3.1 Introduction and least upper bound theorem......Page 101
3.2 Simplifying definitions and proofs of elementary calculus......Page 102
3.3 Continuity and limits for internal functions......Page 106
3.4 More nonstandard characterizations of classical notions......Page 110
3.5 Inverse functions; be......Page 114
3.6 Differentiation......Page 116
3.7 Integration......Page 120
3.8 Pitfalls in nonstandard analysis......Page 123
4.1 Principles of permanence......Page 127
4.2 The saturation principle......Page 132
4.3 Stirling's formula......Page 134
4.4 Nonstandard mathematics without the axiom of choice?......Page 136
Appendix......Page 141
References......Page 143
Index......Page 145