Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
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
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
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
176
Liczba stron
65
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CLXXVI
Daty
wydano
1980
Twórcy
autor
Bibliografia
- [1] R. Chuaqui, Forcing for the impredicative theory of classes, J. Symb. Logic 37 (1972), pp. 1-18.
- [2] W. Easton, Powers of regular cardinals, Ph. D. Thesis, Princeton University, Princeton, N. J., 1964.
- [3] K. Gödel, The consistency of the continuum hypothesis, Annals of Mathematical Studies, Princeton, N. J., 1940.
- [4] J. L. Kelley, General Topology, Van Nostrand, Princeton, N. J., 1955.
- [5] G. Kreisel and A. Lévy, Reflection principles and their use for establishing the complexity of axiomatic sytems, Z. Math. Logik Grundlagen Math. 14 (1968), pp. 97-142.
- [6] K. Kuratowski and A. Mostowski, Set Theory, North Holland, 1968.
- [7] W. Marek, On the metamathematics of the impredicative set theory, Dissertationes Math. 98 (1973).
- [8] W. Marek and P. Zbierski, Axioms of choice in impredicative set theory, Bull. Acad. Polon. Sci. 20 (1972), pp. 255-258.
- [9] R. Montague, Semantical closure and non-finite axiomatizability I. Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics in Warsaw 1959, pp. 45-69, Państwowe Wydawnictwo Naukowe, Warsaw 1961.
- [10] A. Morse, A theory of sets, Academic Press, New York 1965.
- [11] A. Mostowski, Constructible sets with applications, North Holland, Amsterdam 1969.
- [12] J. Myhill and D. Scott, Ordinal definability, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part I, American Mathematical Society, Providence, R. I., 1971, pp. 271-278.
- [13] J. Shoenfield, Unramified forcing, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R. I., 1971, pp. 357-382.
- [14] J. Shoenfield, Mathematical Logic, Addison-Wesley 1967.
- [15] R. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), pp. 1-56.
- [16] A. Tarski, A lattice-theoretical fixed point theorem and Us applications Pacific J. Math. 5 (1955), pp. 285-309.
- [17] L. Tharp, Consistency and independence of GCH for Morse's set theory, Ph. D. Thesis M. I. T. (1965).
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