The axiom of choice, fixed point theorems, and inductive ordered sets
HTML articles powered by AMS MathViewer
- by Milan R. Tasković PDF
- Proc. Amer. Math. Soc. 116 (1992), 897-904 Request permission
Abstract:
This paper continues the study of the inductiveness of posets in terms of fixed apexes and points. The author proves some new equivalents of the Axiom of Choice, i.e., Zorn’s lemma. These statements are of fixed apex type and fixed point type theorems. The paper includes comments about these theorems and presents new characterizations of inductiveness and quasi-inductiveness of posets in terms of fixed apexes and fixed points.References
-
A. Abian, A fixed point theorem equivalent to the axiom of choice, Abstracts Amer. Math. Soc. 4 (1983), no. 388.
- Garrett Birkhoff, Lattice Theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, New York, N. Y., 1948. MR 0029876
- Nicolas Bourbaki, Sur le théorème de Zorn, Arch. Math. (Basel) 2 (1949/50), 434–437 (1951) (French). MR 47739, DOI 10.1007/BF02036949
- Kurt Gödel, The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, No. 3, Princeton University Press, Princeton, N. J., 1940. MR 0002514, DOI 10.1515/9781400881635
- Thomas J. Jech, The axiom of choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0396271 C. Kuratowski, Une methode d’elimination des nombers transfinies raisonnements mathematiques, Fund. Math. 3 (1922), 76-108.
- Djuro Kurepa, Sur la relation d’inclusion et l’axiome de choix de Zermelo, Bull. Soc. Math. France 80 (1952), 225–232 (French). MR 56048, DOI 10.24033/bsmf.1431
- Andrzej Mostowski, On the principle of dependent choices, Fund. Math. 35 (1948), 127–130. MR 29956, DOI 10.4064/fm-35-1-127-130
- Gregory H. Moore, Zermelo’s axiom of choice, Studies in the History of Mathematics and Physical Sciences, vol. 8, Springer-Verlag, New York, 1982. Its origins, development, and influence. MR 679315, DOI 10.1007/978-1-4613-9478-5 H. Rubin, On a problem of Kurepa concerning the axiom of choice, Notices Amer. Math. Soc. 5 (1958), no. 378.
- Herman Rubin and Jean E. Rubin, Equivalents of the axiom of choice, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam-London, 1970. MR 0434812 —, Some new forms of the axiom of choice, Notices Amer. Math. Soc. 7 (1960), no. 380.
- Herman Rubin and Jean E. Rubin, Equivalents of the axiom of choice. II, Studies in Logic and the Foundations of Mathematics, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. MR 798475 M. Tasković, Banach’s mappings of fixed points on spaces and ordered sets, Thesis, Math. Balkanica 8 (1978), 150.
- Milan R. Tasković, On an equivalent of the axiom of choice and its applications, Math. Japon. 31 (1986), no. 6, 979–991. MR 870984
- Milan R. Tasković, Characterizations of inductive posets with applications, Proc. Amer. Math. Soc. 104 (1988), no. 2, 650–659. MR 962843, DOI 10.1090/S0002-9939-1988-0962843-X E. Zermelo, Neuer Beweis fur die Moglichkeit einer Wohlordnung, Math. Ann. 15 (1908), 107-128.
- Max Zorn, A remark on method in transfinite algebra, Bull. Amer. Math. Soc. 41 (1935), no. 10, 667–670. MR 1563165, DOI 10.1090/S0002-9904-1935-06166-X
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 897-904
- MSC: Primary 04A25; Secondary 06A06
- DOI: https://doi.org/10.1090/S0002-9939-1992-1111224-8
- MathSciNet review: 1111224