Natural numerosities of sets of tuples

Forti, Marco; Morana Roccasalvo, Giuseppe

doi:10.1090/S0002-9947-2014-06136-9

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 $\mathbb {N}$.

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