《COMPLETE THEORIES》

Ⅰ．GENERAL INTRODUCTION1

1．1．Preliminary remarks1

1．2．The lower predicate calculus3

1．3．Structures，models，diagrams5

1．4．Godel＇s theorem and its extensions7

1．5．Completeness9

Ⅱ．MODEL-COMPLETENESS12

2．1．Persistence12

2．2．Model-completeness13

2．3．The test for model-completeness15

2．4．Predicates in model-complete theories20

2．5．Partial completeness25

2．6．Related sets28

Ⅲ．MODEL-COMPLETE GROUPS AND FIELDS32

3．1．Abelian groups32

3．2．Algebraically closed fields38

3．3．Real-closed fields43

3．4．Fields with valuation47

3．5．Integral domains with valuation62

3．6．Modules(groups with operators)67

Ⅳ．COMPLETENESS72

4．1．Prime models72

4．2．Joint extensions74

4．3．Complete groups and fields77

4．4．Adjunction of indeterminates82

Ⅴ．APPLICATIONS90

5．1．Direct applications90

5．2．Transfer principles92

5．3．Specialisation of parameters95

5．4．Predicates in algebraically closed fields98

Ⅵ．SYNTACTICAL TRANSFORMS107

6．1．Introduction107

6．2．The bounding transform109

6．3．A counter-example116

6．4．Bounding transforms and deducibility119

6．5．Application124

BIBLIOGRAPHY126

INDEX128