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Internal and forcing models for the impredicative theory of classes

Seria
Rozprawy Matematyczne tom/nr w serii: 176 wydano: 1980
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Warianty tytułu
Abstrakty
EN
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
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
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).
Języki publikacji
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Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-d4a6e003-8f2e-4082-9a9d-d839617f08e8
Identyfikatory
ISBN
83-01-01112-2
ISSN
0012-3802
Kolekcja
DML-PL
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