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I've been checking several Italian books of logic and math to find some expression equivalent to "range over" as in the following mathematical expression:

Let x, y, and z be variables that range over the natural numbers.

I haven't found an equivalent expression so far, but this is the best I could come up with:

Siano x, y e z variabili che vanno sui numeri naturali.

Siano x, y e z variabili che spaziano tra i numeri naturali.

What would be a better equivalent for this?

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    ... appartenenti all'insieme dei numeri naturali (books.google.es/…)? Ma forse si potrebbe riscrivere la frase in modo più semplice, qualcosa come "Siano le variabili x, y e z numeri naturali" (non sono per niente sicura che questa frase suoni bene alle orecchie di un italiano; infatti, neanche a me suona del tutto bene).
    – Charo ♦
    Dec 11, 2021 at 10:30
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    Attenzione @GiuseppeRomanazzi, i naturali non sono un corpo, non sono neppure un anello, quindi non si può usare il termine "campo" in questo contesto (che, peraltro, è un sostantivo e no un verbo come "range over").
    – Charo ♦
    Dec 11, 2021 at 13:03
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    I'd say neither proposal is common usage. Possibly “Siano x, y, z variabili che denotano numeri naturali” is better; perhaps also ”Siano x, y, z variabili nei numeri naturali”.
    – egreg ♦
    Dec 11, 2021 at 14:53
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    Sì, @GiuseppeRomanazzi, i reali e anche i complessi sono un "campo", cioè, un "corpo commutativo", ma i naturali non lo sono.
    – Charo ♦
    Dec 11, 2021 at 17:43
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    I'd also say something like “Siano x, y e z variabili che variano sui/nei numeri naturali”, which is not too elegant due to the repetition variabile/variare but expresses better the “range over” idea, rather than just x, y e z being elements of N.
    – DaG
    Dec 11, 2021 at 18:03

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Appartengono is the equivalent. Range over means that they are inside the specified range and we use 'appartenere' when we refer to something included inside a set of elements (in math). Anyways, it's more likely that you'll find something like that in an Italian book:

Siano x, y, z numeri naturali.

Rarely you'll find something like:

Siano x, y, z appartenenti all'insieme dei numeri naturali.

Unless you are in a section that's talking about set theory

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    Sorry, but I fully disagree: the question is about logic, where variables belong to nothing at all.
    – egreg ♦
    Dec 16, 2021 at 7:21
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    @egreg Are you sure the question really belongs strictly only to logic? It is tagged mathematics and states just that OP is actually looking in "several Italian books of logic and mathematics" for the equivalent of... I'm afraid it might depend on the context. Maybe it is the case to ask the OP for some more explanation. (Just by reading - as it is - the question I would also have said variabili che appartengono ai numeri naturali... Since you obviously see more, when you can find some time could you explain it better?)
    – Hastur
    Dec 16, 2021 at 7:58
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    I was thinking the same
    – Lorenzo Martini
    Dec 16, 2021 at 8:08
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    The problem I find in this proposal is that you are not saying that x, y and z are variables that take values over N.
    – Charo ♦
    Dec 16, 2021 at 13:34
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    @LorenzoMartini The word “logic” appears in the initial paragraph of the question. In mathematical logic the word “variable” has a very specific technical meaning and you can't generally omit it, because the distinction between “variable” and “term” is fundamental.
    – egreg ♦
    Dec 16, 2021 at 16:27

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