Natural numerosities of sets of tuples
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- by Marco Forti and Giuseppe Morana Roccasalvo PDF
- Trans. Amer. Math. Soc. 367 (2015), 275-292 Request permission
Abstract:
We consider a notion of “numerosity” for sets of tuples of natural numbers that satisfies the five common notions of Euclid’s Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show that, contrasting to cardinal arithmetic, the natural “Cantorian” definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (“gauge”) ideal. In particular, special numerosities, called “natural”, can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of $\mathbb {N}$.References
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Additional Information
- Marco Forti
- Affiliation: Dipartimento di Matematica, University of Pisa, Via Buonarroti 1C, 56100 Pisa, Italy
- Email: forti@dma.unipi.it
- Giuseppe Morana Roccasalvo
- Affiliation: Dipartimento di Matematica, University of Pisa, Via Buonarroti 1C, 56100 Pisa, Italy
- Email: moranaroccasalvo@mail.dm.unipi.it
- Received by editor(s): June 18, 2012
- Received by editor(s) in revised form: December 2, 2012
- Published electronically: July 2, 2014
- Additional Notes: The first author’s research was partially supported by MIUR Grant PRIN 2009, Italy.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 275-292
- MSC (2010): Primary 03E65, 03F25; Secondary 03A05, 03C20
- DOI: https://doi.org/10.1090/S0002-9947-2014-06136-9
- MathSciNet review: 3271261