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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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.
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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