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What is an infinite relation and how can it be specified?

What is an infinite relation and how can it be specified? (Brisk · Chapter 1) — "Sets can be infinite, and relations can be too. An infinite relation is simply…

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Les relations peuvent être infinies.
Les ensembles et les relations peuvent contenir une infinité d’éléments.
Une relation infinie contient une infinité de couples ordonnés.
Une relation infinie peut être spécifiée sans énumérer tous ses couples.
On peut la définir intentionnellement, par la signification de son nom.
On peut aussi utiliser une règle qui calcule systématiquement son extension.
Exemple : la relation « isGreaterThan » entre Z et Z.
Z désigne l’ensemble des entiers.
La relation contient des couples comme (5, 2) et (17, –13).
Elle ne contient pas des couples comme (7, 9) ou (11, 11).
La signification de « isGreaterThan » permet de comprendre la relation sans lister tous les couples.

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What is an infinite relation and how can it be specified?

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Brisk

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"Sets can be infinite, and relations can be too. An infinite relation is simply a relation with infinitely many ordered pairs in it. This might seem strange at first, since how could we ever hope to specify all the ordered pairs? But it’s really no different than with sets: we either have to do it intensionally, or else have a rule for systematically computing the extension. As an example of the first, consider the relation “isGreaterThan” between Z and Z. (Recall that “Z” is just a way of writing “the set of integers.”) This relation contains ordered pairs like (5, 2) and (17, –13), since 5 isGreaterThan 2 and 17 isGreaterThan –13, but not (7, 9) or (11, 11). Clearly it’s an infinite relation. We couldn’t list all the pairs, but we don’t need to, since the name implies the underlying meaning of the relation."

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