Numerosities of point sets over the real line
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- by Mauro Di Nasso and Marco Forti PDF
- Trans. Amer. Math. Soc. 362 (2010), 5355-5371 Request permission
Abstract:
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. $\textbf {cov}(\mathcal {B})=\mathfrak {c}< \aleph _{\omega }$), 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.References
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Additional Information
- Mauro Di Nasso
- Affiliation: Dipartimento di Matematica “L. Tonelli”, Università di Pisa, Pisa, Italy
- MR Author ID: 610241
- Email: dinasso@dm.unipi.it
- Marco Forti
- Affiliation: Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, Pisa, Italy
- Email: forti@dma.unipi.it
- Received by editor(s): July 25, 2008
- Published electronically: May 19, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 5355-5371
- MSC (2010): Primary 03E65, 03E05; Secondary 03E35, 03A05, 03C20
- DOI: https://doi.org/10.1090/S0002-9947-2010-04919-0
- MathSciNet review: 2657683