## Notre Dame Journal of Formal Logic

### Partition Principles and Infinite Sums of Cardinal Numbers

Masasi Higasikawa

#### Abstract

The Axiom of Choice implies the Partition Principle and the existence, uniqueness, and monotonicity of (possibly infinite) sums of cardinal numbers. We establish several deductive relations among those principles and their variants: the monotonicity follows from the existence plus uniqueness; the uniqueness implies the Partition Principle; the Weak Partition Principle is strictly stronger than the Well-Ordered Choice.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 36, Number 3 (1995), 425-434.

Dates
First available in Project Euclid: 17 December 2002

https://projecteuclid.org/euclid.ndjfl/1040149358

Digital Object Identifier
doi:10.1305/ndjfl/1040149358

Mathematical Reviews number (MathSciNet)
MR1351415

Zentralblatt MATH identifier
0843.03027

#### Citation

Higasikawa, Masasi. Partition Principles and Infinite Sums of Cardinal Numbers. Notre Dame J. Formal Logic 36 (1995), no. 3, 425--434. doi:10.1305/ndjfl/1040149358. https://projecteuclid.org/euclid.ndjfl/1040149358

#### References

• 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 MR 91k:03128
• Fillmore, P. A., “An Archimedean property of cardinal algebras,” Michigan Journal of Mathematics, vol. 11 (1964), pp. 365–367. Zbl 0192.09604 MR 29:5757
• 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. Zbl 0223.02055 MR 42:2933
• Häussler, A. F., “Defining cardinal addition by $\leq$-formulas,” Fundamenta Mathematicæ, vol. 115 (1983), pp. 195–205. MR 85e:03111
• 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 MR 93e:03072
• 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 MR 94j:03106
• Jech, T. J., The Axiom of Choice, North-Holland, Amsterdam, 1973. Zbl 0259.02051 MR 53:139
• König, D., “Zur Theorie der Mächtigkeiten,” Rendiconti Circolo Matematico di Palermo, vol. 26 (1908), pp. 339–342.
• Moore, G. H., Zermelo's Axiom of Choice, Springer-Verlag, New York, 1982. Zbl 0497.01005 MR 85b:01036
• 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 MR 80k:03053
• Pincus, D., “Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods,” The Journal of Symbolic Logic, vol. 37 (1972), pp. 721–743. Zbl 0268.02043 MR 49:2374
• Pincus, D., “Cardinal representatives,” Israel Journal of Mathematics, vol. 18 (1974), pp. 321–343. Zbl 0302.02021 MR 51:2913
• Rubin, H., and J. E. Rubin, Equivalents of the Axiom of Choice. II, North-Holland, Amsterdam, 1985. Zbl 0582.03033 MR 87c:04004
• Sageev, G., “An independence result concerning the Axiom of Choice,” Annals of Mathematical Logic, vol. 8 (1975), pp. 1–184. Zbl 0306.02060 MR 51:2915
• Sierpiński, W., “Sur une proposition qui entraî ne l'existence des ensembles non mesurables,” Fundamenta Mathematicæ, vol. 34 (1947), pp. 157–162. Zbl 0038.03203 MR 9,338i
• Tarski, A., “Cancellation laws in the arithmetic of cardinals,” Fundamenta Mathematicæ, vol. 36 (1949), pp. 77–92. Zbl 0039.04804 MR 11,335b
• Tarski, A., Cardinal algebras, Oxford University Press, New York, 1949. Zbl 0041.34502 MR 10,686f
• Zermelo, E., “Neuer Beweis für die Möglichkeit einer Wohlordung,” Mathematische Annalen, vol. 65 (1908), pp. 107–128.