CONTENTS

Introduction............................................................................................................ 5

I. Axiom system and elementary consequences........................................... 6
1. Axioms........................................................................................................................ 6
2. Definitions and elementary consequences........................................................ 9

II. Principles of definitions by recursion........................................................... 12
3. Monotone operations.............................................................................................. 12
4. Recursion principles............................................................................................... 13
5. The rank function...................................................................................................... 15

III. Well-ordering relations................................................................................... 16
6. Well-ordering relations............................................................................................ 19
7. Ordinals and -well-order types.............................................................................. 20

IV. Models and satisfaction................................................................................. 25
8. Satisfaction................................................................................................................ 25
9. Absoluteness............................................................................................................ 29
10. Models of MT........................................................................................................... 31

V. The axiom of constructibility........................................................................... 32
11. The axiom of constructibility................................................................................. 32

VI. Ordined definability......................................................................................... 37
12. Existence of hierarchies of all classes and relative constructibility............. 37
13. Ordinal definability................................................................................................. 39

VII. Complexity of the axiom system.................................................................. 43
14. Non-finite axiomatizability and non-axiomatizability with sentences of bounded unrestricted quantifier depth.................. 43
15. Complexity of axioms for the predicative sentences....................................... 46

VIII. Forcing models............................................................................................. 47
10. Forcing..................................................................................................................... 47
17. Products of notions of forcing and coherent notions...................................... 55

IX. Independence of axioms of choico.............................................................. 59
18. Automorphisms of notions of forcing................................................................. 59
19. Independence of the local axiom of choico....................................................... 61
20. Independence of the global axiom of choico.................................................... 62

References............................................................................................................ 65