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Partition principles and infinite sums of cardinal numbers. (English) Zbl 0843.03027

The author investigates weak forms of the axiom of choice (AC) which are related to the partition principle. (PP: The injective and the surjective orderings of cardinals coincide.) For example, there is a weakening of PP which is equivalent to ACW (choice for well-ordered families of sets.) It is still unknown, if PP is equivalent to AC. An interesting contribution towards this problem is the following Theorem 3.6: PP follows from the assertion that the cardinality of a sum of a family of cardinals does not depend on the chosen representatives.
Reviewer: N.Brunner (Wien)

MSC:

03E25 Axiom of choice and related propositions
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[1] Banaschewski, B., and G. H. Moore, “The dual Cantor-Bernstein theorem and the Partition Principle,” Notre Dame Journal of Formal Logic , vol. 31 (1990), pp. 375–381. · Zbl 0716.03044 · doi:10.1305/ndjfl/1093635502
[2] Fillmore, P. A., “An Archimedean property of cardinal algebras,” Michigan Journal of Mathematics , vol. 11 (1964), pp. 365–367. · Zbl 0192.09604 · doi:10.1307/mmj/1028999191
[3] Halpern, J. D., and P. E. Howard, “Cardinals \(m\) such that \(2m=m\),” Proceedings of the American Mathematical Society , vol. 26 (1970), pp. 487–490. JSTOR: · Zbl 0223.02055 · doi:10.2307/2037365
[4] Häussler, A. F., “Defining cardinal addition by \(\leq\)-formulas,” Fundamenta Mathematic æ, vol. 115 (1983), pp. 195–205.
[5] Howard, P. E., “The Axiom of Choice for countable collections of countable sets does not imply the Countable Union Theorem,” Notre Dame Journal of Formal Logic , vol. 33 (1992), pp. 236–243. · Zbl 0760.03014 · doi:10.1305/ndjfl/1093636102
[6] Howard, P. E., “Unions of well-ordered sets,” Journal of the Australian Mathematical Society , Series A, vol. 56 (1994), pp. 117–124. · Zbl 0797.03048
[7] Jech, T. J., The Axiom of Choice , North-Holland, Amsterdam, 1973. · Zbl 0259.02051
[8] König, D., “Zur Theorie der Mächtigkeiten,” Rendiconti Circolo Matematico di Pa lermo, vol. 26 (1908), pp. 339–342. · JFM 39.0099.02
[9] Moore, G. H., Zermelo ’s Axiom of Choice, Springer-Verlag, New York, 1982. · Zbl 0497.01005
[10] Pelc, A., “On some weak forms of the Axiom of Choice in set theory,” Bulletin de l ’Académie Polonaise des Sciences, Série des Sciences Mathématique, Astronomique et Physiques, vol. 26 (1978), pp. 585–589. · Zbl 0434.03029
[11] Pincus, D., “Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods,” The Journal of Symbolic Logic , vol. 37 (1972), pp. 721–743. JSTOR: · Zbl 0268.02043 · doi:10.2307/2272420
[12] Pincus, D., “Cardinal representatives,” Israel Journal of Mathematics , vol. 18 (1974), pp. 321–343. · Zbl 0302.02021 · doi:10.1007/BF02760841
[13] Rubin, H., and J. E. Rubin, Equivalents of the Axiom of Choice . II, North-Holland, Amsterdam, 1985. · Zbl 0582.03033
[14] Sageev, G., “An independence result concerning the Axiom of Choice,” Annals of Mathematical Logic , vol. 8 (1975), pp. 1–184. · Zbl 0306.02060 · doi:10.1016/0003-4843(75)90002-9
[15] Sierpiński, W., “Sur une proposition qui entraî ne l’existence des ensembles non mesurables,” Fundamenta Mathematic æ, vol. 34 (1947), pp. 157–162. · Zbl 0038.03203
[16] Tarski, A., “Cancellation laws in the arithmetic of cardinals,” Fundamenta Mathematic æ, vol. 36 (1949), pp. 77–92. · Zbl 0039.04804
[17] Tarski, A., Cardinal algebras , Oxford University Press, New York, 1949. · Zbl 0041.34502
[18] Zermelo, E., “Neuer Beweis für die Möglichkeit einer Wohlordung,” Mathematische Annalen , vol. 65 (1908), pp. 107–128. · JFM 38.0096.02
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