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Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction..................................................................................................................................................5
I. Topological forcing..................................................................................................................................8
1.1. Preliminaries.......................................................................................................................................8
1.2. Languages, satisfaction, and relative constructibility..................................................................10
1.3. Forcing..................................................................................................................................................15
1.4. Adding a set of generic sets.............................................................................................................26
1.5. Adding a proper class of generic sets...........................................................................................26
1.6. Collapsing cardinals.........................................................................................................................33
II. Reflection principles and their consequences................................................................................39
2.1. Reflection principles..........................................................................................................................39
2.2. The Tree and Ramsey properties...................................................................................................46
III. Application to relative consistency results.......................................................................................50
3.1. The main result...................................................................................................................................50
3.2. A converse consistency result..........................................................................................................54
References..................................................................................................................................................58
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
91
Liczba stron
60
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom XCI
Daty
wydano
1972
Twórcy
autor
Bibliografia
- [1] P. Bernays, Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, Essays on the Foundations of Mathematics, Jerusalem 1961, pp. 3-49.
- [2] P. J. Cohen, Set Theory and the Continuum Hypothesis, New York 1966.
- [3] W. Easton, Powers of regular cardinals. Annals of Math. Logic. 1 (1970), pp. 139-178.
- [4] P. Erdös and A. Tarski, On some problems involving inaccessible cardinals, Essays on the Foundations of Mathematics, Jerusalem 1961, pp. 50-82.
- [5] H. Gaifman, Measurable cardinals and constructible sets, Amer. Math. Soc. Notices, 11 (1964), p. 771.
- [6] K. Gödel, Consistency-proof for the generalized continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A., 25 (1939), pp. 220-224.
- [7] K. Gödel, The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the Axioms of Set Theory, fifth printing, Princeton 1961.
- [8] A. Hajnal, On a consistency theorem connected with the generalized continuum problem, Acta Math. Acad. Sci. Hung., 12 (1961), pp. 321-376.
- [9] W. Hanf and D. Scott, Classifying inaccessible cardinals, Amer. Math. Soc. Notices, 8 (1961), p. 445.
- [10] H. J. Keisler and A. Tarski, From accessible to inaccessible cardinals. Fund. Math. 53 (1964), pp. 225-308.
- [11] K. Kuratowski, Topology I, New York 1966.
- [12] A. Lévy, Axiom schemata of strong infinity in axiomatic set theory. Pacific J. Math., 10 (1960), pp. 223-238.
- [13] A. Lévy, Definability in axiomatic set theory-1, Logic, Methodology and Philosophy of Science, Proc. of the 1964 International Congress, Amsterdam 1965, pp. 127-151.
- [14] A. Lévy, A generalization of Gödel's notion of constructibility, J. Symb. Logic 25 (1960), pp. 147-155.
- [15] Th. A. Linden, Equivalences between Gödel's definitions of constructibility, Sets, Models and Recursion Theory, Proc. of the Summer School in Mathematical Logic and Tenth Logic Colloquium, Amsterdam 1967, pp. 33-43.
- [16] G. Mackey, Equivalence of a problem in measure theory to a problem in the theory of vector lattices, Bull. Amer. Math. Soc. 50 (1944), pp. 719-722.
- [17] A. Mostowski, Some impredicative definitions in the axiomatic set-theory, Fund. Math. 37 (1950), pp. 111-124.
- [18] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., Ser. 2, 30 (1928), pp. 338-384.
- [19] W. Reinhardt, Ackerman's set theory equals ZF, Annals of Math. Logic 2 (1970), pp. 189-248.
- [20] R. M. Robinson, Finite sequences of classes, J. Symb. Logic 10 (1945), pp. 125-126.
- [21] F. Rowbottom, Some strong axioms of infinity incompatible with the axiom of constructibility, Annals of Math. Logic 3 (1971) pp. 1-44.
- [22] D. Scott, Axiomatizing set theory, mimeograph lecture notes, 1967 Summer Institute on Axiomatic Set Theory, Univ. of California, Los Angeles.
- [23] D. Scott, Measurable cardinals and constructible sets, Bull. Acad. Polon. Sci., Ser. des Sci. Math., Astr. et Phys., 7 (1961), pp. 145-149.
- [24] J. Shepherdson, Inner models for set theory I, J. Symb. Logic 16 (1951), pp. 161-190.
- [25] J. R. Shoenfield, Mathematical Logic, Reading, Mass., 1967.
- [26] J. R. Shoenfield, On the independence of the axiom of constructibility, Amer. J. of Math. 81 (1959), pp. 537-540.
- [27] J. Silver, Some applications of model theory in set theory, Annals of Math. Logic 3 (1970), pp. 45-110.
- [28] R. Solovay, A $Δ^1_3$ non-constructible set of integers, Trans. Amer. Math. Soc., 127 (1967), pp. 50-75.
- [29] G. Takeuti, On the axiom of constructibility, mimeograph, Univ. of Illinois, 1966.
- [30] G. Takeuti, Topological space and forcing, mimeograph, Univ. of Illinois, 1966.
- [31] A. Tarski and R. Vaught, Arithmetical extensions of relational systems, Composito Math. 13 (1957), pp. 81-102.
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