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Choice principles in Węglorz’ models

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Węglorz' models are models for set theory without the axiom of choice. Each one is determined by an atomic Boolean algebra. Here the algebraic properties of the Boolean algebra are compared to the set theoretic properties of the model.
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  • Department of Mathematics, Universität für Bodenkultur, Gregor-Mendel-Straße 33, A-1180 Wien, Austria , brunner@edv1.boku.ac.at
  • Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395, U.S.A., jer@math.purdue.edu
Bibliografia
  • [1] J. Aczél, Bemerkungen zur Realisierung der Hausdorffschen Axiome in abstrakten Mengen, Publ. Math. Debrecen 3 (1953), 183-186.[2] M. Boffa, Arithmetic and the theory of types, J. Symbolic Logic 49 (1984), 621-624.[3] N. Brunner, Dedekind-Endlichkeit und Wohlordenbarkeit, Monatsh. Math. 94 (1982), 9-31.[4] N. Brunner, Amorphe Potenzen kompakter Räume, Arch. Math. Logik 24 (1984), 119-135.[5] N. Brunner, Linear operators and Dedekind-sets, Math. Japon. 31 (1986), 1-16.[6] N. Brunner, The Fraenkel-Mostowski method, revisited, Notre Dame J. Formal Logic 31 (1990), 64-75.[7] N. Brunner, 75 years of independence proofs by Fraenkel-Mostowski permutation models, Math. Japon. 43 (1996), 177-199.[8] N. Brunner, K. Svozil and M. Baaz, Effective quantum observables, Nuovo Cimento B 110 (1995), 1397-1413.[9] J. W. Degen, There can be a permutation which is not the product of two reflections, Z. Math. Logik Grundlag. Math. 34 (1988), 65-66.[10] N. Dunford and J. T. Schwartz, Linear Operators I, Interscience, New York, 1957.[11] S. Feferman, Some applications of the notions of forcing and generic sets, Fund. Math. 56 (1965), 325-345.[12] G. Grätzer, General Lattice Theory, Birkhäuser, Basel, 1978.[13] J. D. Halpern, On a question of Tarski and a theorem of Kurepa, Pacific J. Math. 41 (1972), 111-121.[14] W. Hodges, Six impossible rings, J. Algebra 31 (1974), 218-244.[15] P. Howard and J. E. Rubin, The weak forms of the axiom of choice package", e-print, http://www.math.purdue/~jer.[16] T. Jech, The Axiom of Choice, North-Holland, Amsterdam, 1973.[17] P. T. Johnstone, Stone Spaces, Cambridge Univ. Press, Cambridge, 1986.[18] S. Koppelberg, General theory of Boolean algebras, in: J. D. Monk and R. Bonnet (eds.), Handbook of Boolean Algebras, North-Holland, Amsterdam, 1989, 1-312.[19] P. A. Loeb, A new proof of the Tychonoff theorem, Amer. Math. Monthly 72 (1965), 711-717.[20] A. Mostowski, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip, Fund. Math. 32 (1939), 201-252.[21] D. Pincus, The strength of the Hahn-Banach theorem, in: Proc. the 1972 Victoria Symposium on Nonstandard Analysis, Lecture Notes in Math. 369, Springer, Heidelberg, 1973, 203-248.[22] J. B. Remmel, Recursive Boolean algebras, in: J. D. Monk and R. Bonnet (eds.), Handbook of Boolean Algebras, North-Holland, Amsterdam, 1989, 1097-1165.[23] H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice II, North-Holland, Amsterdam, 1985.[24] D. Schmeidler, Cores of exact games I, J. Math. Anal. Appl. 40 (1972), 214-225.[25] R. Sikorski, Boolean Algebras, Springer, Heidelberg, 1969.[26] J. K. Truss, Classes of Dedekind finite cardinals, Fund. Math. 84 (1974), 187-208.[27] J. K. Truss, The axiom of choice for linearly ordered families, Fund. Math. 99 (1978), 133-139.[28] B. Węglorz, A model of set theory $\hbox\ninegot S$ over a given Boolean algebra, Bull. Acad. Polon. Sci. 17 (1969), 201-202.
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