Di Nasso, Mauro; Forti, Marco Numerosities of point sets over the real line. (English) Zbl 1200.03033 Trans. Am. Math. Soc. 362, No. 10, 5355-5371 (2010). Summary: We consider the possibility of a notion of size for point sets, i.e. subsets of the Euclidean spaces \(\mathbb{E}_{d}(\mathbb{R})\) of all \(d\)-tuples of real numbers, that satisfies the fifth common notion of Euclid’s Elements: “the whole is larger than the part”. Clearly, such a notion of “numerosity” can agree with cardinality only for finite sets. We show that “numerosities” can be assigned to every point set in such a way that the natural Cantorian definitions of the arithmetical operations provide a very good algebraic structure. Contrasting with cardinal arithmetic, numerosities can be taken as (nonnegative) elements of a discretely ordered ring, where sums and products correspond to disjoint unions and Cartesian products, respectively. Actually, our numerosities form suitable semirings of hyperintegers of nonstandard analysis. Under mild set-theoretic hypotheses (e.g. cov\((\mathcal{B})=\mathfrak{c}< \aleph _{\omega })\), we can also have the natural ordering property that, given any two countable point sets, one is equinumerous to a subset of the other. Extending this property to uncountable sets seems to be a difficult problem. Cited in 7 Documents MSC: 03E10 Ordinal and cardinal numbers 03C20 Ultraproducts and related constructions 03E05 Other combinatorial set theory 03H05 Nonstandard models in mathematics Keywords:size of a point set; fifth common notion of Euclid’s Elements; numerosity; cardinality; arithmetical operations; semirings of hyperintegers; nonstandard analysis; natural ordering PDFBibTeX XMLCite \textit{M. Di Nasso} and \textit{M. Forti}, Trans. Am. Math. Soc. 362, No. 10, 5355--5371 (2010; Zbl 1200.03033) Full Text: DOI References: [1] Vieri Benci and Mauro Di Nasso, Numerosities of labelled sets: a new way of counting, Adv. Math. 173 (2003), no. 1, 50 – 67. · Zbl 1028.03042 [2] Vieri Benci, Mauro Di Nasso, and Marco Forti, An Aristotelian notion of size, Ann. Pure Appl. Logic 143 (2006), no. 1-3, 43 – 53. · Zbl 1114.03055 [3] Vieri Benci, Mauro Di Nasso, and Marco Forti, An Euclidean measure of size for mathematical universes, Logique et Anal. (N.S.) 50 (2007), no. 197, 43 – 62. · Zbl 1131.03029 [4] A. BLASS - Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory , to appear. · Zbl 1198.03058 [5] A. BLASS, M. DI NASSO, M. FORTI - Quasi-selective ultrafilters and countable numerosities, in preparation. · Zbl 1270.03105 [6] G. CANTOR - Mitteilungen zur Lehre vom Transfiniten, Zeitschr. Philos. philos. Kritik 91 (1887), 81-125; 92 (1888), 240-265. [7] Georg Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 49 (1897), no. 2, 207 – 246 (German). · JFM 28.0061.08 [8] G. CANTOR - Gesammelte Abhandlungen mathematischen und philosophischen Inhalts , Berlin, 1932 (reprinted 1990). · JFM 58.0043.01 [9] C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. · Zbl 0697.03022 [10] Joseph Warren Dauben, Georg Cantor, Princeton University Press, Princeton, NJ, 1990. His mathematics and philosophy of the infinite. · Zbl 0858.01028 [11] Euclid, The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III – IX. Vol. III: Books X – XIII and Appendix, Dover Publications, Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath; 2nd ed. · Zbl 0071.24203 [12] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. · Zbl 1007.03002 [13] Piotr Koszmider, On coherent families of finite-to-one functions, J. Symbolic Logic 58 (1993), no. 1, 128 – 138. · Zbl 0783.03030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.