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Transactions of the American Mathematical Society

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Numerosities of point sets over the real line

Authors: Mauro Di Nasso and Marco Forti
Journal: Trans. Amer. Math. Soc. 362 (2010), 5355-5371
MSC (2010): Primary 03E65, 03E05; Secondary 03E35, 03A05, 03C20
Published electronically: May 19, 2010
MathSciNet review: 2657683
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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

Marco Forti
Affiliation: Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, Pisa, Italy

Received by editor(s): July 25, 2008
Published electronically: May 19, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.